Mix 4 cups (500 g) all-purpose flour, 1 Tbsp. plus 1 tsp. Diamond Crystal or 2¼ tsp. Morton kosher salt, 1 Tbsp. sugar, and one ¼-oz. envelope instant or rapid-rise dry yeast (about 2¼ tsp.) in a large bowl with a rubber spatula to combine. Pour in 1⅔ cups room-temperature water and mix until a sticky dough forms and no dry streaks of flour remaining. Cover tightly with a lid or plate and let sit at room temperature 15 minutes.

Set a small bowl of water on work surface. Uncover dough and, using a wet hand, stretch one side of dough up and over to meet the opposite side; turn bowl 90°. Repeat 3 more times, turning bowl 90° after each. This constitutes 1 full fold. If dough doesn’t always stretch to meet opposite side, just aim for the middle. If dough begins to feel sticky, re-wet your hands and shake off excess, before continuing. Cover bowl and let rest 30 minutes. (Set a timer.) Repeat process two more times, letting dough rest 30 minutes after each full fold for a total of 3 full folds (if you lose count halfway through, just fold 1 more time and proceed). By the end of the third fold, dough should be significantly smoother and tighter.

Generously coat a 13x9" baking pan, preferably metal, with extra-virgin olive oil. Transfer dough to prepared pan, turning to coat on all sides with oil. Turn dough, seam side down. Using your fingers, press dough out until it reaches edges of pan. If dough resists, let rest 5–10 minutes before trying again. Cover pan (a rimmed baking sheet is excellent for this) and let dough rise in a warm spot until puffy, jiggly, and nearly doubled in size, 1–1½ hours, depending on the temperature of your room.

Place a rack in middle of oven; preheat to 425°. Whisk together 3 garlic cloves, finely grated, 2 Tbsp. double-concentrated tomato paste, 2 Tbsp. extra-virgin olive oil, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, 1 tsp. sugar, ¼ tsp. crushed red pepper flakes, and 3 Tbsp. room-temperature water in a small bowl.

Spoon sauce over focaccia, spreading evenly across surface with spoon. Holding your fingers like claws, press down into focaccia to dimple all over. Scatter 12 oz. mixed cherry tomatoes, halved, over, pushing down into dimples.  Bake focaccia until puffed and golden around edges, 20–25 minutes. Reduce oven temperature to 375° and continue to bake until deep golden around edges, 15–20 minutes more. Let focaccia cool in pan 10 minutes, then transfer to a wire rack and let cool completely.

Just before serving, drizzle focaccia with oil and sprinkle with flaky sea salt.