Mix 1/2 cup bread flour, 1/4 cup water, and sourdough discard together with a wooden spoon. Add a little flour to the bowl of a stand mixer. Add starter mixture and allow to rise in a warm area, without direct sunlight, until doubled in size.

Attach dough hook and begin to mix the starter. Add remaining flour, 3 tablespoons of olive oil, and 1 teaspoon salt. Mix in enough remaining water until dough pulls away from sides and forms a ball. Knead dough in the mixer for 5 minutes.

Transfer dough to a work surface and knead by hand for 5 minutes. Score, cover with a cloth, and allow to rise in a draft-free place until doubled in volume, at least 2 hours or up to overnight.

Lightly oil a shallow 12x16-inch baking pan. Knead dough for 5 minutes and spread into the pan. Whisk 1 tablespoon oil and 1 tablespoon water together. Sprinkle lightly over focaccia. Cover with plastic wrap and let rise until doubled.

Preheat the oven to 400°F (200°C).

Press into focaccia with your fingertips, leaving indentations 1 1/2 inches apart. Distribute olives and cherry tomatoes evenly, pressing them into the dough. Sprinkle with rosemary leaves, flaked salt, and the oil-water mixture.