Slice the tomato into ¼ inch thick rounds. Lay tomato slices on paper towels in a single layer. Pat the top side dry with additional paper towels. Let them sit for 2-3 minutes while you prepare the other ingredients. This crucial step removes excess moisture and prevents your bread from getting soggy.

Lay out 4 slices of sourdough bread on a clean work surface. Spread 1 tablespoon of basil pesto evenly on one side of each bread slice. Keep the pesto layer thin and even, about ⅛ inch thick. Don't overspread or the sandwich will be too greasy.

Place 2-3 slices of mozzarella cheese on the pesto side of 2 bread slices. Arrange the dried tomato slices on top of the cheese, covering the surface evenly. Sprinkle a tiny pinch of salt and pepper over the tomatoes. Top each with the remaining bread slices, pesto side facing down. Press down gently to seal the sandwiches together.

Place a large skillet or griddle over medium heat. Add 1 tablespoon of butter and let it melt, swirling to coat the pan evenly. Make sure the heat is medium, not high, to prevent burning the bread before the cheese melts.

Carefully place both sandwiches in the buttered skillet. Cook for 3-4 minutes without moving them, allowing the bottom to develop a golden-brown crust. While the bottom cooks, spread the remaining tablespoon of butter on the top side of the bread slices.

Using a wide spatula, carefully flip each sandwich over. The first side should be crispy and golden brown. Cook the second side for another 3-4 minutes until golden and the cheese is fully melted. Gently press down with the spatula to ensure even contact with the pan and to help the cheese melt.

Remove the sandwiches from heat when both sides are crispy and golden brown. Transfer to a cutting board and let rest for 1 minute. This allows the cheese to set slightly and makes slicing easier without everything sliding out.

Using a sharp knife, cut each sandwich diagonally from corner to corner. This creates two triangular halves that look cafe-style. Serve immediately while hot and crispy with the cheese still melty. Add optional garnishes like fresh basil, balsamic glaze, or red pepper flakes if desired.