Have it your way. Serve this dish hot, warm, or at room temperature. It makes a great side for fish, chicken, or pork.
Place oil and 1 tablespoon of the butter in a large skillet; heat over medium heat until butter melts. Add endive halves cut side down; sprinkle with salt and pepper. Cook until endive begins to lightly caramelize, about 3 minutes. Turn endive halves and cook until lightly browned, 2 to 3 minutes more.
Add broth to skillet and bring to a simmer. Cover and reduce heat to medium-low. Cook until endive is just tender and cores are soft enough to be easily pierced with a paring knife, about 5 minutes, turning once. Uncover and increase heat to medium-high. Add honey and lemon juice; cook until liquid has reduced by half, about 2 minutes. Gently stir in remaining 1/2 tablespoon butter. Serve sauce over endive halves. Sprinkle with fresh nutmeg, if desired.