Heat a large skillet over medium heat and cook the ground beef or turkey until browned, then pour off the grease completely. The meat should lose its raw color and smell savory instead of sharp, and the pan should not have a slick layer of fat left behind.

Stir in the diced onion and cook for 3 to 5 minutes until it softens. The onion should look slightly translucent and smell sweet and mellow rather than harsh.

Add the chopped cabbage, tomatoes with green chilies, tomato sauce, chili powder, garlic powder, salt, and pepper, then stir until the cabbage is coated. The skillet will look very full at first, but the cabbage starts shrinking almost right away as it heats.

Cover the skillet, keep the heat at medium, and cook for 15 to 18 minutes, stirring now and then so nothing catches on the bottom. The cabbage is ready when it turns translucent and fork-tender but still keeps a little shape instead of collapsing.

Scatter the shredded cheese over the top, take the skillet off the heat, and cover it for about 2 minutes so the cheese can melt. The cheese should look glossy and fully softened, with a gooey top layer that settles into the hot cabbage mixture.