Rub cut faces of tomatoes over the large holes of a box grater, using the flattened palm of your hand to move the tomatoes back and forth.
The flesh should be grated off, while the skin remains intact in your hand. Discard skin and season tomato pulp with kosher salt to taste.
Adjust oven rack to just below grill and preheat grill to high.
Place bread, cut side up, on a cutting board and brush with some olive oil. Season with salt.
Place bread, oiled side up, on a rack set in a tray or directly on the grill rack and broil until crisp and starting to char around the edges, 2 to 3 minutes.
Remove bread from oven and rub with the split garlic clove.
Spoon tomato mixture over bread. Drizzle with remaining extra-virgin olive oil and season with flaky sea salt.