Place 2 cups all-purpose flour and 1 teaspoon kosher salt in a large bowl and whisk to combine.
Place 3/4 cup warm water and 1 teaspoon barley malt syrup or honey in a small bowl and whisk to combine. Sprinkle 1 teaspoon active dry yeast over the top. Let stand until the yeast dissolves and is foamy, about 5 minutes.
Add the yeast mixture and 2 tablespoons of the olive oil to the flour mixture and stir with a flexible spatula until no dry flour remains.
Remove the dough from the bowl. Clean and dry the bowl, then coat the bowl with 1 tablespoon of the olive oil. Return the dough to the bowl and turn to coat in the oil. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1 hour; alternatively, let rise in the refrigerator for 12 hours.
Halve 1/2 cup cherry tomatoes. Cut 2 ounces fresh mozzarella cheese into 1/2-inch pieces (about 1/2 cup). Pick the leaves from 2 fresh basil sprigs and coarsely chop until you have 1 tablespoon.
Place 1 tablespoon of the olive oil in a 10-inch cast iron skillet and rub it into the bottom and sides of the pan. Place the dough in the skillet. Gently stretch to the edges of the pan; if the dough is very sticky, use damp hands or a plastic dough scraper. Use your fingertips to gently dimple the surface of the dough.
Arrange the tomatoes, mozzarella, basil, and remaining 1 tablespoon olive oil evenly over the top. Sprinkle with 1/4 teaspoon flaky salt. Cover with plastic wrap and let rise in a warm place until puffed and slightly risen, 45 minutes if the dough was at room temperature or 1 hour if the dough has been refrigerated.
Heat an outdoor grill for direct, medium heat (425° to 450°F).
Place the skillet directly on the grates. Cover and cook until the edges of the focaccia are dark-golden brown, 20 to 25 minutes.
Place the skillet on a wire rack and let cool for 5 minutes. Remove the focaccia from the pan, place on the rack, and let cool completely. Top with more fresh basil if desired. Cut into wedges and serve.