Prepare the grill for indirect cooking over medium heat (350° to 450°F).
On a stovetop or side burner preheat a large skillet over medium heat for a few minutes. Add the oil and tomatoes and stir to coat. Cook over medium heat until the tomatoes begin to break down, 8 to 10 minutes, stirring occasionally. Stir in the remaining ingredients and continue to cook until the tomatoes release their juices and the sauce thickens, 3 to 5 minutes, stirring occasionally. Transfer to a bowl and cool to room temperature.
Mix the patty ingredients, and then gently and briefly form four patties of equal size, each about (1 inch) thick. With your thumb or the back of a spoon, make a shallow indentation about (1 inch) wide in the center of each patty to prevent them from forming a dome as they cook. Refrigerate the patties until ready to grill.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Grill the patties over direct medium heat, with the lid closed, until cooked to 160°F (or your preferred doneness), 8 to 10 minutes for medium, turning once or twice. During the last minute of grilling time, toast the rolls, cut side down, over direct heat.
Serve the patties on the toasted rolls with the jam, lettuce, red onion, and avocado.