Position a rack in the center of the oven; heat the oven to 200°F.
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-inch margin at the edge. Follow with the tomatoes and basil, and sprinkle with 1/4 teaspoon each kosher salt and pepper. Fold the tortillas in half to enclose the filling, creating a half-moon.
In a 10- or 12-inch 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 secondside is golden brown and the cheese has melted completely, about 2 minutes. (Watch carefully as the Parmigiano crust can burn easily; lower the heat if it’s getting too dark.) Transfer the quesadillas to the oven to keep warm and repeat with the remaining two quesadillas. Cut the quesadillas in half (or in smaller wedges if serving as an hors d’oeuvre) and serve.
Note: Water-packed mozzarella is a fresh style and should be bought and consumed within a few days. It doesn’t grate very well, so just cut it into small pieces. Vacuum-sealed mozzarella is usually much drier and can be easily grated.
You can hold the cooked quesadillas in the warm oven for up to 30 minutes.