In a large bowl, combine the ground turkey, breadcrumbs, egg, garlic, Italian seasoning, salt, and pepper. Mix gently with your hands just until combined — overworking makes dense burgers. Divide into 4 equal portions and shape into 4-inch patties, slightly thinner in the center so they cook flat. Wash your hands well after handling raw turkey.

Heat the olive oil in a large heavy skillet over medium-high heat until shimmering. Lay in the patties and cook undisturbed for 5 minutes, until a deep golden crust forms underneath.

Flip with a wide spatula and cook 6–7 minutes more, until the patties are firm and an instant-read thermometer inserted sideways into the thickest part reads 165°F. This is the safety step that matters — turkey must hit 165°F internally. (If the outsides brown too fast, lower the heat to medium.)

Lay a slice of fresh mozzarella on each patty, cover the skillet with a lid or a sheet of foil, and cook 1–2 minutes, until the cheese melts and drapes over the patty. Transfer the patties to a plate.

Spread the cut sides of the brioche buns with the softened butter. Toast cut-side down in the same hot skillet, 1–2 minutes, until golden and crisp. Watch them — brioche browns fast.

Place each patty on a toasted bottom bun. Top with a tomato slice, a small handful of basil leaves, and a zigzag of balsamic glaze. Cap with the top bun and serve immediately with fries or a side salad.