3 tablespoons unsalted butter
2 pounds onions, sliced into thin half-moons
Kosher salt and freshly ground black pepper
4 garlic cloves, sliced
2 small sprigs thyme, leaves only
4-inch sprig rosemary
2 cups chicken broth, divided
1 tablespoon balsamic vinegar
2 tablespoons Dijon mustard
3 pounds boneless skinless chicken thighs
2 ounces Gruyère cheese, finely grated or shaved (about 1 cup)