In a small mixing bowl, combine the diced tomatoes, minced garlic, fresh basil, olive oil, salt, and pepper. Toss to combine and let sit for 10 minutes to allow flavors to meld.
Spread the softened butter evenly across one side of each of the four bread slices.
Place two slices of bread, butter-side down, into a cold non-stick skillet or onto a griddle.
Layer 1/2 cup of shredded mozzarella onto each of the bread slices in the pan.
Using a slotted spoon to avoid excess moisture, place a generous portion of the tomato mixture over the cheese. Drizzle the balsamic glaze over the tomato layer.
Top with the remaining mozzarella cheese and place the remaining bread slices on top, butter-side facing up.
Turn the heat to medium-low. Cook the sandwiches for 4 to 5 minutes on the first side until the bread is golden brown and the bottom layer of cheese begins to melt.
Carefully flip the sandwiches using a wide spatula. Cook for another 3 to 4 minutes until the second side is toasted and the cheese is fully melted and internal temperature is reached.
Remove from the skillet and allow the sandwiches to rest for 1 minute before slicing diagonally to serve.