James,
Yes. If I'm intent on the cover reseeding I just mow and let the residue lay. If I'm ready to prepare for planting, I will follow up the mowing with disking.
Field Peas are especially well suited to this technique. One double, double row patch provided 4 generations of green manure and soil building this summer , seed, mow, re-grow, mow, re-grow, mow, regrow. That patch is now planted in my onion crop which will be harvested next May and will also produce my 2021 potato crop.
In addition to field peas, I also use alfalfa for summer cover and mow it every two weeks until frost. This summer I made 8 cuttings of alfalfa this way providing incalculable valuable soil building. Just mow it with the lawn mower and let it lay. Right now, I have my winter covers going and will mow and then disc those next March for the spring planting. My winter crops include broc, cabbage, brussels, carrots, turnips, radishes, kale, chard, collards, and beets. These crops flourish without artificial fertilizers or insecticides in space that was previously covered with legumes.
It has always baffled me as to why the use of cover crops, summer and winter, isn't much more wide spread. It is so easy and so extremely productive.