Nothing really new to add here, just thought this process might be useful. It might not apply to your climate, where I am we have super wet springs.
Last growing season I covered grass with inoculated burlap sacks (requires a little foresight, I set up the inoculation for the bags about 5 weeks before I used them). Covered the burlap sacks with "weeds" when I had them.
Covered those sacks with anywhere from 3 to 12 inches of wood chips.
In those wood chips I made soil plugs, to the depth of the grass, and maybe 6 inches in diameter. The soil was a mix of big-box-store bagged soil and purchased compost.
Planted in soil plugs.
Burlap sacks--free from a local roaster (check whole foods stores, coops, etc, bulk grains often come in giant brown paper bags).
Bagged soil: 1.79 per 40 lb bag. Bought 10
Delivered compost: $50 per cubic yard bought 1
Mushroom spawn: 20 per #5 bag from Fedco (I wanted known spawn of edible mushrooms, maybe try store bought mushroom slurries?)
Total cost: 90 bucks for about 1000 square feet of bed space, but again I only planted into those "soil plugs". Took about a week to go from grass to planted beds, working 2-3 hours a day.
Squash, sunflowers, various herbs, bare root strawberries, lilies, scarlett runner beans, all flourished for me. All brassica lagged, they were planted too deep into summer and not watered enough maybe. Cucumbers looked promising until they wilted to cucumber beetle pressure.