In a food processor, combine the basil with the almonds and garlic and pulse until the almonds are finely chopped. Add the 1/2 cup of olive oil and process to a paste. Add the grated Pecorino and pulse to combine. Season the pesto with salt and pepper. You should have about 1 cup.
Transfer 1/4 cup of the pesto to a bowl. Add the ground chuck and ground sirloin and a pinch of salt and gently knead to blend. Form the mixture into six 4-inch patties, about 3/4 inch thick. Brush the burger patties lightly with olive oil.
Light a grill and oil the grates. Grill the burgers over moderately high heat for 3 minutes. Flip the burgers, top them with the mozzarella slices and close the grill. Cook for 3 minutes longer for medium-rare burgers. Grill the buns until lightly toasted, then spread some of the pesto on the bottoms. Top with the burgers, the tomatoes and the remaining pesto. Close the burgers and serve right away.