Preheat the oven to 400°F. Arrange the tomatoes, cut-side down, on a layer of paper towels to absorb liquid for 30 minutes. Roll out the puff pastry between two pieces of parchment paper to about 10 x 12 inches in size. Peel off the top later of parchment and place the pastry on a baking sheet. Pierce the pastry with a fork all over. Using a sharp paring knife, score the pastry to create a border ½ inch from the edges of the pastry. Brush the pastry with the 1/2 cup pesto, then scatter the pastry with the tomatoes, goat cheese, and herbs. Season with salt and pepper to taste. Bake until the tomatoes soften and the pastry is puffed and golden, 18-20 minutes. Remove from the oven and, while still warm, scatter with the Parmigiano Reggiano and additional herbs, if desired.