Adjust rack to top position and preheat oven to 425 degrees. (TIP: For extra-crispy potatoes, place a baking sheet inside oven while it preheats.) Wash and dry all produce. Cut potatoes into 1/2-inch-thick wedges. Halve, peel, and very thinly slice onion. Finely dice tomato. Halve buns.

Toss potatoes on a baking sheet with a large drizzle of oil and Italian Seasoning; season generously with salt and pepper. (If you preheated your baking sheet, toss potatoes in a large bowl, then place on sheet.) Arrange potatoes in a single layer. Roast on top rack until golden brown and crisp, 20-25 minutes.

Meanwhile, heat a large drizzle of oil in a large pan over medium-high heat. Add onion and cook, stirring occasionally, until softened, 4-5 minutes. Stir in 1/4 cup water and cook until liquid has evaporated, 2-3 minutes. Add tomato, vinegar, and 1 tsp sugar (2 tsp for 4 servings); reduce heat to mediumlow. Cook, stirring occasionally, until very soft, 8-10 minutes.

While jam cooks, form sausage into two 1/2-inch-thick patties (four patties for 4 servings), each slightly wider than a burger bun. TIP: Wet hands to prevent meat from sticking.

Stir 1 TBSP butter (2 TBSP for 4 servings) into jam until melted. Season with salt and pepper; transfer to a medium bowl. Wash out pan. Heat a drizzle of oil in same pan over medium-high heat. Add patties and cook until browned and cooked through, 4-7 minutes per side. In the last 2 minutes of cooking, top each patty with mozzarella. Cover pan until cheese is melted.

Toast buns until golden brown. Fill toasted buns with patties and jam. Serve with potato wedges on the side.