In a large bowl, mix the flour, yeast, and salt. Add the water, honey, and olive oil. Stir with a wooden spoon or Danish whisk until combined—no kneading needed. Cover with plastic wrap and let rest for 1 hour at room temperature.
Wet your fingers and fold the dough over itself 6–8 times right in the bowl. Cover and rest for 30 minutes. Repeat this process 3 more times: fold 4 times, rest 30 minutes, and cover. Total: 4 rounds of folding.
Drizzle the dough with olive oil, cover, and refrigerate for 12–24 hours.
Grease a 9x13-inch (22x33 cm) tray with plenty of olive oil and line with parchment paper. Transfer the dough to the tray, fold it once or twice, and place seam-side down. Cover and let it rise at room temperature until doubled.
Preheat your oven to 450°F (230°C) with a pizza stone or baking steel inside for at least 30 minutes.
Dimple the dough using wet fingers, spreading it gently to fill the tray. Press the cherry tomato halves into the dough, then scatter the rosemary and shallots on top.
Mix leftover rosemary-infused oil with 2 tbsp of water. Whisk to emulsify and drizzle over the entire surface. Finish with a generous sprinkle of coarse salt.
Place the tray on the hot pizza stone and bake for 25–30 minutes, until golden and crisp. If the top browns too fast, tent with foil or place another tray above it.
Transfer to a wire rack and let cool for 15 minutes before serving.