Turn the turkey fillet through a large meat grinder grate or chop into minced meat with a knife.
Peel the onion and garlic, finely chop and sauté in butter until soft, 7 to 10 minutes. Puree with a blender. Peel the potatoes and grate on the smallest grater.
Mix turkey, potatoes and onion-garlic puree. Season with salt, pepper and Provence herbs. Form 4 round cutlets 2 cm thick, put on a parchment-lined board and refrigerate for 30-60 minutes.
Cut the rolls in half lengthwise and toast immediately over hot coals on the side of the cut. Brush the cuts with pesto.
Preheat a wire rack, brush with oil, add the burgers and grill, turning 2 times, until cooked through, about 8 minutes. Reheat the roasted vegetables at the same time.
Arrange vegetables and meatballs on buns and serve immediately.