Pre-heat oven to 325ºF.
Arrange tomato slices on a screen set over a sheet pan and drizzle with about 2 tablespoons EVOO, some salt and pepper. Roast the tomatoes until tender and caramelized, about an hour. Reserve.
When the tomatoes are ready, place a large skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. While the pan is heating up, combine the sirloin with some salt and pepper, and form into 4 patties. Cook the patties in the hot skillet until seared and cooked through, about 6 minutes per side.
While the patties are cooking, place a small pot over medium-high heat with the butter. Add the garlic to the melted butter and cook for about 1 minute. Add the flour to the pan and cook for about 1 minute. Whisk the milk into the butter-flour mixture and bring up to a bubble. Simmer until the sauce has thickened, about 2 minutes. Season with salt, lots of pepper (about 1 teaspoon) and stir in the cheese. Reserve the sauce warm.
When the burgers are ready, assemble them by placing some lettuce on the bun bottoms and topping it with a burger patty, some Parmigiano sauce, a couple of roasted tomatoes, some basil and balsamic drizzle.