Heat grill to medium-high. In a medium bowl, place pork, minced onion, 1/4 cup barbecue sauce, 1 teaspoon coarse salt, and 1/4 teaspoon pepper. Mix very gently with a fork to combine, being careful not to overwork meat (which would make burgers dense).
Gently form meat into 4 equal-size patties, 4 inches in diameter and 1 inch thick. With thumb, make a 1/2-inch-deep indentation in the center of each (this helps keep burgers flat during grilling).
Moisten a folded paper towel with oil; grasp with tongs and rub over grates. Season burgers with salt and pepper; grill until cooked throughout, 5 to 8 minutes per side.
Serve on onion rolls (toasted on grill, if you like), with sliced onion and remaining barbecue sauce.
You can substitute ground beef chuck or ground turkey in this recipe. Vary cooking times accordingly.