Can we cook on the 1st day of the 7th Biblical Month known as Yom Teruah, the Day of Shouting/Blowing of Trumpets, Remembrance of Blowing of Trumpets?

Answer From The Annual Reading Schedule - Week 28 Notes For Leviticus: No. In Leviticus 23:24, we are told to have a Shabbatown (rest) on the 1st Day of the 7th Biblical Month (Yom Teruah / Day of Shouting/Blowing). Also, since we're commanded to have a Set-Apart Gathering and Reading on this day, this day would be considered a Set-Apart Day which inherently means all the laws and regulations of the weekly Sabbath would apply. See the answer to the similar question for Feast of Weeks which elaborates more on a "Set-Apart Day".

Also, as discussed in my answer for the 1st and 7th Days of the Feast of Unleavened Bread, I do not view Exodus 12:16 as transferrable or applicable to any other feast days outside of the Feast of Unleavened Bread. See the answer to the question for the 1st and 7th Days of Unleavened Bread for the longer explanation on this.

In addition, I'd like to point out the events in Nehemiah 8. In Nehemiah 8:1-2 we learn they were reading from the Torah on the 1st Day of the 7th Biblical Month which is Yom Teruah. In Nehemiah 8:9, they say, "Hay'Yowm Qadosh-Huw La'Yahweh" (phonetically, Hay'Yom Qadosh-Hu La'Yahuweh) which means:

The Day Set-Apart-He To'Yahuweh" or "The Day is Set-Apart to Yahuweh

You see, this day is "Set-Apart" to Yahweh. The question for us is – is this day Set-Apart to us? Will we treat it like a Set-Apart Day? What does it mean for us if a day is "Set-Apart to Yahweh"?

We can see a few things from Nehemiah 8 showing what this means.

We can see that the ideal behavior on any Set-Apart Day is to not mourn or weep. We learn this from Nehemiah 8:9.

We can also see that preparing our food in advance is also part of observing a Set-Apart Day and guarding it properly. We learn this from Nehemiah 8:10 which states that they were to send portions to those who had not prepared. Why wouldn't people "be prepared"? Because the people at that time did not know until that day when they were actually in the process of reading the law that the Day was Set-Apart and therefore they could not cook on it. This is part of the reason why I think the people were so emotionally impacted at the reading because they realized that they had not prepared for that day as they should have. It's quite possible, some of them had already violated the Set-Apartness of that day by cooking on it since the evening the night before and were weeping when they realized the sin they had committed that day because they had not prepared in advance and had cooked on it before hearing the Torah. There they were, learning something they did not know before, that the very day in which they were hearing the words was a Set-Apart Day. It seems they were cut to their heart over their sin and sorry over their sin because they realized how they had not guarded what is Set-Apart to Yahweh. And so they were told in Nehemiah 8:9 to not mourn or weep. And they were told in Nehemiah 8:10 to send portions to those for whom none is prepared!

Why were they told to send portions to those for whom none is prepared? Was it because those who didn't have any prepared were poor? No. It is because they could not cook on this day and so many of them were probably not prepared with food cooked in advance because they didn't know they had to prepare until the day had already arrived. In fact, it tells you right in Nehemiah 8:10 the reason why:

send portions to those for whom none is prepared, for this day is set-apart…

The reason why portions had to be sent to those for whom none is prepared is not because "there are lots of poor people without". Of course, on any feast day, everyone is expected to share with the poor among them, as they are able, so that the poor also rejoice with them as well. However, directly in verse 10 it clearly says the reason why is "for this day is Set-Apart". It is because the Day is Set-Apart to Yahweh that they had to send portions for whom none is prepared – because on a day Set-Apart to Yahweh we should have prepared and done our cooking in advance as we learn from the first day Set-Apart – the weekly 7th Day Sabbath.

Also, if you research the word "prepared" in Nehemiah 8:10, we find it is from the Hebrew word linked to the Strong's # H3559 – "kun". This is the same Strong's # for the word "prepare" in Exodus 16:5 which is saying that they will prepare their food on the 6th day of the week in preparation for the weekly Sabbath! This looks very obvious to me that the word in Exodus 16:5 is about cooking food in advance of a Set-Apart Day as part of keeping the day Set-Apart in a way that Yahweh desires. I don't think it's a huge stretch to say that this is also very clearly required on other Set-Apart Days. Nehemiah 8:10 is confirmation of this to me. The reason why they had to send portions to those for whom none is prepared is because we are not permitted to cook on Set-Apart days. We need to be prepared in advance.

And, maybe, there is an additional reason behind this. Perhaps one day some catastrophic events will occur on days of rest and if we are not prepared in advance we may not have the food and drinks we desire to rejoice on those days instead of mourn or weep because of the destruction or sadness around us from the world. There is a lesson in Nehemiah 8:10 for us to learn – some of them were not prepared. We do not want to be in that same situation. We want to be prepared.