Dissolve sugar and yeast in 1 cup warm water in a large bowl; let stand 5 minutes. Stir in fine sea salt. Lightly spoon flour into dry measuring cups and spoons; level with a knife. Add 2 cups plus 2 tablespoons flour, stirring to form a soft dough. Turn dough out onto a floured surface. Knead dough until smooth and elastic (about 8 minutes); add enough of remaining 1/3 cup flour, 1 tablespoon at a time, to prevent dough from sticking to hands.
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 45 minutes or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.)
Heat oil in a small skillet over medium-low heat. Add garlic; cook 5 minutes or until fragrant. Remove garlic from oil with a slotted spoon; discard garlic, and remove pan from heat.
Place dough on a baking sheet coated with cooking spray; pat into a 12 x 8-inch rectangle. Brush garlic oil over dough; sprinkle with thyme. Cover and let rise 25 minutes or until doubled in size.
Preheat oven to 425°.
Make indentations in top of dough using the handle of a wooden spoon or your fingertips; sprinkle dough evenly with coarse sea salt. Bake at 425° for 14 minutes or until lightly browned. Remove from pan; cool on a wire rack.