Heat 2 teaspoons oil in a nonstick skillet over medium-high heat. Add onion, 1 tablespoon honey, and thyme; cook 5 minutes or until browned, stirring occasionally. Remove from heat; stir in vinegar and 1/4 teaspoon salt. Cool completely.
Dissolve yeast and remaining 1 tablespoon honey in 1 1/4 cups warm water in a large bowl; let stand 5 minutes. Lightly spoon flour into dry measuring cups; level with a knife. Add onion mixture, remaining 2 tablespoons oil, remaining 1 1/4 teaspoons salt, and 3 1/2 cups flour to yeast mixture; stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes). Add enough of remaining 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, 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes. Gently press dough into a 15 x 10-inch jelly-roll pan coated with cooking spray. Cover and let rise 25 minutes or until almost doubled in size.
Preheat oven to 400°.
Sprinkle dough with cheese. Bake at 400° for 18 minutes or until browned. Cool in pan on a wire rack 10 minutes. Remove from pan; cool completely on rack.