Are dinner reservations for 2 people needed 1st week of June?

It really depends on the venue. Cafés don't require, or probably even take, reservations. I always make dinner reservations, as eating is one of my pleasures in Paris and I enjoy researching.

If you enjoy eating early, you might consider Atelier Maitre Albert in the 5th, just across from Notre Dame. It's one of a handful of fine restaurants that opens at 6:30. All the meats are spit-roasted and are delicious. I'd definitely reserve.

And keep in mind that you can reserve on line at a lot, if not most, restaurants these days. Please make sure to cancel if you can't make a standing reservation. It's rude not to.

That’s “my” neighborhood. There are a number of eateries in the area that are “service continu” so no problem if you still have a 5 or 6 PM stomach. Eating at that time you will not need reservations unless you do fine dining. I just got home today after having spent 3 weeks in the area and often ate at 6 with no problem. They will generally ask if you are “eating or drinking”, lol.

Even small local restaurants that are well regarded often book up especially on weekend evenings. Cafes and brasseries with continuous service are more walk in -- but restaurants which usually open at 7 or 7:30? if you want to eat at the good ones, yes reserve. Famous plays weeks in advance; local places often the day before will do.

If you want to dine before 19h30, select cafés, brasseries, crêperies, or somewhere specifying "service continue" meaning they do not close between 14h and 19h as do restaurants.

If you go to a restaurant, reserve.

I do not often dine near École Militaire, but consider these restaurants:

Milagro
https://www.milagroparis.com/carte

Auberge Bressane
https://www.auberge-bressane.com/

Café Max
https://www.cafemax.fr/assets/pdf/CARTE-DES-METS.pdf

D'Chez Eux
https://www.chezeux.com/#reserver

At a restaurant, everyone is expected to order a main course. You can share a dessert if you like.

I would echo Tocard's recommendation of Auberge Bressane. I've enjoyed several meals there in the past.

If you want to be sure that you can eat where you want to eat, reservations are paramount.

Another vote for Bressane. they have yummy dessert souffles

Dining in Paris is not like other cities. If a place in early June is not booked by 7:30-8:00 dinner time, it is probably not any good. If you want to dine earlier than than that, confine yourself to places identified as a bouillon, a brasserie, or a cafe, where there is continuous service throughout the day. Any place calling itself a restaurant is unlikely to take you at 7:00 or before.

We too will be in Paris the first week in June. I already have reservations for my first and last night. When I went to make reservations at two places I ran into something I haven’t seen before. At two $$-$$$ places they were wanting my credit card for a 20 euro downpayment if we didn’t show for the reservation. Not happy with that!

CAJan, it's becoming more and more common as people do not cancel a dinner reservation and then don't show up. I find it hard to blame the restaurants.