In a small skillet, heat oil and butter over medium heat. Add onion, salt and sugar; cook and stir until onion is softened, 3-4 minutes. Reduce heat to medium-low; cook until deep golden brown, stirring occasionally, 4-6 minutes.
In a large bowl, mix jelly, salt and pepper. Add beef; mix lightly but thoroughly. Shape into four 1/2-in.-thick patties.
Grill burgers, covered, over medium heat or broil 4 in. from heat until a thermometer reads 160°, 4-5 minutes on each side. Serve on buns with caramelized onion, cheese and, if desired, jalapeno slices.