Arrange a rack in center of oven; preheat to 425°.
On a cutting board, arrange cabbage on its side. Using a sharp knife, cut 1" off top of cabbage (reserve for another use). Cut remaining cabbage into 4 (½"-thick) steaks; discard root end.
Arrange steaks on a baking sheet. Brush both sides of steaks with oil; season all over with 1 ½ tsp. kosher salt and a few grinds of pepper.
Roast cabbage until tender, 20 to 25 minutes.
About 5 minutes before cabbage is finished cooking, form beef into 4 large flat patties about 4" in diameter; season all over with remaining 1 tsp. salt and a few grinds of pepper. Arrange one patty on top of each roasted cabbage steak.
Continue to bake until cabbage is tender and burgers are cooked to desired degree of doneness (about 8 minutes for medium-rare).
Top each patty with a slice of cheese. Continue to bake just to melt cheese, about 1 minute.
Serve topped with ketchup, mustard, bacon, pickles, and onion, as desired.
