You can plant them a little deep, which will help protect the roots from this week's hot temperatures. Mulching the ground will also keep the roots cooler. Young plants can snap in half pretty easily so transplant carefully.

I don't know how large your plants are or whether they've been raised indoors or out, but they need to be fully hardened off to full sun and hot temperatures before they go into the ground or they won't survive, so if you raised them indoors or in mostly shade, be sure to complete the proper process of hardening them off before you transplant them. The weather this week is going to be really hard on them.

I started them indoors with seed starter soil. Broccoli too. But the peas are really skinny and tall and beginning to fall over. I was afraid the hot weather this week would be brutal on them.
I have prepared a spot for them by adding a little compost to the soil. Should I set the little seedlings close to that bed, still in their tray? Or start off in the shade and gradually move them to the (almost) "full sun" bed? How gradually would you move them? A little more sun each day for a week?

I chose the peas and broccoli from the OSU fall guide...and the seed packets said these seeds could be started/planted in August thru September. It's so hot though.
Would it be better to put some seeds directly into the ground? Is it less harsh on the young plants if they start that way?
Thank you.

You don't say what kind of peas. Southern or English Shelling or Sugarsnap? Although I'm not sure it makes a difference. I would definitely start some in the ground, as even cool weather peas like Sugarsnaps like to germinate in warm rather than cool soil. Not hot, just warm, so I would lightly mulch and keep the soil damp. Let us know how they do.