You could plant part of it in, say, fava beans & then cut them off leaving the roots etc in the ground then plant your veg crop in spring as usual the beans will have fertilized the soil & the tops will be good mulch or compost material. Why dont you just plant Fall/Winter veg. & let that be the cover crop-in, at least, part of the beds? (You cant eat your cover crop if you need food). Then cut off the spent plants, leave roots in the ground, fertilize as usual & plant your Spring/Summer crops over them. You could do what I'm going to do, plant the cover crops in the paths, cut them off in the spring & use the dry matter for mulch/compost material. You really dont need to tie all the bed space in a cover crop. What is your reasoning? If you have your soil structure developed you DONT want to be tilling as this will destroy what nature has been improving. Plant some alfalfa seed around a few spots, it will root down deep, fertilize & loosen the soil without tearing up everything then chop the tops off a while before U want to use the spot for veg. I'll get you straightened out!

plant whatever you want and turn it under with a shovel or a fork...I plant potatoes and as soon as they are planted I throw in handfulls of seed...whatever. kale, spinach carrots whatever I have lots of...I turn them under while I am hilling potatoes and then if any is left I eat them as they grow and just end up scratching the remnants into the soil.

lonmower

You can certainly plant green manure in your beds. It shouldn't be tilled, and it doesn't even have to be turned unless you'd rather turn the tops under, too. Leaving the nutrient-rich roots in the soil is what provides your soil with plant goodies. The easiest way is to just cut the plants close to ground level, and and in the spring plant your garden among the roots. Use the tops of the green manure for mulch, as others have said.

If you have lots of compost, I would weedwhack the cover crop before it goes to seed, then cover everything with compost early enough to start breaking it down.

Or, same as above, but save the weedwhacked cover crop to mulch on top of the compost.

Growing stuff in roots of a previous cover crop isn't always fun/easy...depending what the cover crop is, it's sometimes hard to transplant thru, and can be difficult to start seeds in. If the crop was a type of legume like peas, leave the roots in instead of pulling the plant. If something like winter rye, I would whack it down and at least fork it under, unless you can add, say, 4-6" of compost to act as a planting zone above it.

I've grown cover crops of both winter rye as well as crimson clover in the past and don't know which was more a problem... the winter rye's roots were strong like grass, hard to break up, etc, and the clover was almost a perennial & hard to keep from regrowing, impossible to pull the roots, plus irresistible to groundhogs.