Preheat the oven to 400°F. Grease a 2-quart (7 X 11-inch) baking dish.
In a large, deep skillet, heat the olive oil over medium-high heat. Add the ground beef, onion, salt, and pepper. Cook until the beef is browned, about 5 minutes, stirring and breaking up the meat with a wooden spoon.
Add the garlic, paprika, thyme, and cayenne and cook for 30 seconds, stirring.
Add the drained tomatoes and chopped cabbage. Continue to cook, stirring often, until the cabbage is tender, about 5 minutes.
Turn the heat off and stir in one cup of the shredded cheese.
Using a slotted spoon, transfer the mixture to the prepared baking dish. Sprinkle it with the remaining cheese.
Bake, uncovered, until the casserole is bubbly and the cheese is browned, 20-30 minutes.
Let the casserole rest for 10 minutes before serving.
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Notes
You want the cabbage roughly chopped, so it's best to buy a whole head of cabbage and chop it yourself. Bagged chopped cabbage is usually finely chopped.
Instead of cheddar, you can use other melty cheeses such as provolone. Shredded mozzarella also works, but its flavor profile is milder.
When the casserole is ready, some liquid will be on the bottom. After the casserole has cooled off a little, you can carefully tip the baking dish into the sink and drain some of it off. But it's OK to just leave it in the pan.
You can keep the leftovers in an airtight container in the fridge for 3-4 days. Reheat them in the microwave, covered, at 50% power. Like many casseroles, it improves after being refrigerated for a day or two. You can also freeze this casserole for up to three months.