Position a rack in the center of the oven; heat to 200 degrees.
In a small bowl, mix the butter and Parmigiano. Spread this on one side of each tortilla and set the tortillas on a work surface, buttered side down. Distribute the mozzarella among the tortillas, covering only half of each and leaving a 1"margin at the edge. Follow with the tomatoes and basil, and sprinkle with 1/4 teaspoons each kosher salt and pepper. Fold the tortillas in half to enclose the fillinf, creating a half moon.
In a 10-12" nonstick skillet over medium heat, cook two of the quesadillas, covered, until golden brown on the first side, about 3 minutes. Uncover, flip, and cook until the second side is golden brown and cheese has melted completely. Lower heat if Parmigiano crust is getting to dark. Transfer quesadillas to the oven to keep warm and repeat with the remaining two quesadillas. You can hold in oven up to 30 minutes.
Cut the quesadillas in half or in smaller wedges if serving as appetizers and serve.