Pour the warm water (about 110°F) into a large mixing bowl, stir in the sugar until dissolved, then sprinkle the instant yeast over the top. Let sit 5 to 10 minutes until foamy and yeasty.
Add the bread flour, fine sea salt, and 2 tablespoons of olive oil to the foamy yeast mixture. Stir with a dough whisk or wooden spoon until no dry flour remains; the dough will be shaggy, wet, and sticky. Scrape down the sides of the bowl. There is no kneading in this recipe.
Drizzle 1 tablespoon of olive oil into a clean large bowl, scrape the dough into it, and turn once to coat. Cover with plastic wrap or a damp towel and let rise in a warm spot for about 1 hour, until doubled in size and bubbly.
Warm the 1/3 cup of olive oil in a small saucepan over medium-low heat for about 1 minute, until warm and shimmery but never smoking. Remove from the heat and stir in the minced garlic and chopped rosemary for 30 to 60 seconds. Set aside to cool, spooning about 2 tablespoons into a small bowl for brushing after baking.
Spoon a scant 1 teaspoon of olive oil into each of the 12 muffin cups, using about 3 tablespoons total, and swirl or tilt the pan to coat the bottoms and sides.
Drizzle 1 tablespoon of olive oil on your work surface and oil your hands lightly. Gently scrape the risen dough onto the counter, pat into a rough rectangle, and cut into 12 equal pieces (about 2 1/2 ounces or 70 g each). Tuck the edges of each piece underneath to form a loose ball and place seam side down in a muffin cup.
Cover the pan with a clean towel and let the muffins rise for 30 to 45 minutes, until puffy and nearly reaching the top of the cups. About 20 minutes before the rise finishes, preheat the oven to 400°F with a rack in the center.
Oil your fingertips and press straight down into each muffin to the bottom of the cup, making 2 to 3 deep dimples per muffin. Spoon the garlic rosemary oil over the muffins (about 2 teaspoons each, reserving the 2 tablespoons set aside) and sprinkle evenly with flaky sea salt.
Bake on the center rack for 18 to 20 minutes, rotating the pan halfway through, until deeply golden with crisp edges and an instant-read thermometer reads about 200°F in the center.
Brush the hot muffins with the reserved garlic rosemary oil. Cool in the pan for 5 minutes, run a small knife around each edge to loosen, then move to a wire rack and serve warm.