You can add minced garlic the last minute or two of cooking time. It takes very little time to saute garlic. I prefer it minced because it tastes much better, and I dislike getting a large piece of garlic in my mouth. I hope this is helpful to you.
I do a weird saute technique with a tiny big of vegetable stock, about two to three tablespoons, rather than oil and add the garlic to that, and then the vegetables. It works well with veggies that you don't want to brown, like broccolli. You have to stir it and watch it like a hawk, if the water completely evaporates, the veggies take on a nasty sweet-burned taste.
I love to add sauteed garlic or shallots to veggies, especially the green ones. What I always do is gently sautee garlic and/or shallots in a small amount of oil w/butter for a VERY short time. Then set aside in a little Pyrex type dish. I then cook the veggies in the usual manner w/water. When cooked, drain off the water and add the sauteed garlic/shallots. Mix well and gently reheat. Add S&P as desired. I like to do this quite a bit in advance of dinnertime as the veggies will really absorb the flavor of the garlic and/or shallots. Easy and really adds a nice kick to plain boiled veggies.
Norma's method is the best if you don't want a mouthful of garlic (I'm one of those who loves it, but this is what I do when I have guests I think might not like it). Otherwise, toss in the garlic at the same time you do the vegs. I rarely use garlic without using onion, and I toss in the garlic AFTER the onions to avoid burning the garlic. There are actually some dishes where you want browned crisp garlic, but generally you don't, and putting something else in the frying pan first avoids that problem. Another thing you can do is nuke a clove or two for only a minute or two, then chop and you can add it later in the dish because it becomes SO mild. Good hint for garlic wimps.
I saute mine at the beginning and I do not remove them. Just make sure your heat is not too high. After cooking for a short while add other ingredients to the pan. The additional food will keep the garlic from burning. Just make sure to give it a good stir so the garlic is not still on the bottom of the pan.
The best way to infuse the taste of ANY aromatic is to start in cool or room temp. oil, broth or water. And LOW heat. See Tip
In your case, have your oil and WHOLE peeled cloves of garlic, in a pan, on low heat. Toast it for about 5-8 mins until the garlic is a light/med tan color. Take out garlic, add in veggies, cook to desired doneness.BTW the garlic would be great schmeered on some french bread.
Tip.........all aromatics [veggies, herbs, spices ect] have oils, when they are drawn out slowly as I mentioned you will get the fullest flavor possible.