Heat oil in a skillet over medium-high heat. Cook chorizo and bacon for 3 to 5 min. or until bacon is crispy. Reduce heat to medium, add onion and cook 3 min., stirring frequently. Add garlic and peppers and cook 15 min., stirring frequently, until peppers start to soften. Add tomatoes, salt and pepper. Simmer for 7 to 10 min., stirring frequently, until liquid has almost evaporated. Keep warm.
Preheat barbecue to medium-high. Form veal into 4 equal-size patties. Season with salt and pepper. Grill patties for 4 to 5 min. per side or until cooked through. In the last 2 min. of cooking, place a slice of cheese on each burger to melt.
Divide patties among buns, top each with 2 tbsp (30 mL) piperade and 2 basil leaves. Serve immediately. Refrigerate leftover piperade in a resealable jar for 6 days or freeze up to 3 months.