Prep the Bread and Heat the Grill: Slice your ciabatta roll in half lengthwise (or lay out your sourdough slices). Preheat your panini press or grill pan over medium heat so it's hot and ready when you begin assembling.
Brush with Olive Oil: Using a basting brush or the back of a spoon, coat the outside of the bottom slice of bread with 1 tablespoon of olive oil. This will help create that beautifully crisp, golden crust when grilled.
Assemble the Layers: Place the oiled side down on the hot grill surface. On the exposed side, layer the fresh mozzarella, followed by the tomato slices and fresh basil leaves. Drizzle the balsamic vinegar evenly over the top. Season lightly with kosher salt and black pepper to taste.
Finish the Sandwich: Brush the outer side of the top slice of bread with the remaining tablespoon of olive oil and place it over the sandwich. Close the lid of the panini press. If using a grill pan, press the sandwich with a heavy skillet or spatula to flatten slightly and grill evenly.
Grill to Golden Perfection: Grill the sandwich until the outside is crisp and golden brown, and the mozzarella begins to melt around the edges. This usually takes about 3–4 minutes in a press or 2–3 minutes per side in a grill pan. If using a pan, flip once halfway through.
Serve and Enjoy: Remove from the heat, let cool slightly, then slice in half and serve warm while the cheese is still melty and the bread is perfectly toasted.