Step 1

Position rack in upper third of oven. Preheat to 450°F.

Step 2

Toss onions, oil, butter, sage (or thyme), salt and pepper in a large bowl. Spread in a 9-by-13-inch metal baking pan (don't use glass or ceramic, which might shatter). Roast, turning once, until browned on both sides, 30 to 35 minutes total.

Step 3

Turn the onions again and carefully add broth to the pan. Continue roasting until the broth is reduced and the onions are very soft, 15 to 20 minutes more.