Josh, that is a bit question and depends on the climate, soil types and fertility, and water availability. If you have good texture and fertile soil, a diverse grass area (not a monoculture of one or two grass species), not too many competitive weedy species in your grass, and a source of irrigation, I would do a combination of heavy mulches and cover crops in different sections of your grass area to build up the soil without disturbing the microbial community already there. I would test your soil and choose a cover crop mix based on your soil nutrient and texture needs. Then I would mow and lightly till-in your cover crops while also leaving un-tilled, undisturbed heavily mulched areas near the tilled areas. Eventually, you can also choose to leave some of your cover crop mix in areas as living mulches and or ground cover for perennials. It is worth doing the homework to know your soil and site very well before you choose cover crops, living mulches, and soil amendments. Good luck building your soil organic matter and microbial community.