Heat the oven to 400 degrees . On a baking sheet, toss the tomatoes and shallots in 1 tablespoon EVOO; season with salt and pepper. Roast for 15 minutes; set aside.
Meanwhile, combine the ground beef, parsley and garlic and form into 4 patties, pressing your thumb into the center of each burger to prevent it from bulging when cooking. In a large nonstick skillet, heat 1 tablespoon EVOO, 1 turn of the pan, over medium-high heat and cook the burgers for 7 to 8 minutes on each side for well-done.
While the burgers cook, in a medium saucepan, melt the butter over medium heat, whisk in the flour and cook for 1 minute. Whisk in the milk and cook until thickened, 3 to 4 minutes. Add the provolone and, using a wooden spoon, stir in a figure-eight motion until combined; season with salt and pepper.
Toss the arugula with the lemon juice, remaining 1 tablespoon EVOO and salt and pepper to taste. Add the warm tomatoes and shallots and toss to combine.
Serve each burger with a pile of salad and a small bowl of cheese sauce for dipping.