Preheat the oven to 400° and line a large baking sheet with parchment paper. In a medium bowl, combine the Parmigiano-Reggiano cheese with the butter, flour and a generous pinch of pepper and mash the ingredients until a dough forms. Form the dough into a 4-inch log. Cut the log into 12 equal slices and arrange the slices on the baking sheet. Using your fingers, press the slices into 1 1/2-inch rounds.
Bake the tuiles on the lowest rack in the oven for about 7 minutes, rotating the pan halfway through baking, until the tuiles are golden brown and sizzling. Let the tuiles cool completely, then transfer them to a paper towel–lined plate and blot the excess fat.
In a small bowl, toss the tomatoes with the chives, tarragon and olive oil and season with salt and pepper. Spoon the salad over the tuiles and serve right away.