Sauté sliced onions and minced garlic in the 1 tablespoon of butter and 1 tablespoon olive oil over medium heat. Cook them until brown and caramelized, about 15 to 20 minutes. Allow to cool for a few minutes.
Pour warm water into a bowl. Sprinkle yeast on top. Add the honey and 3 tablespoons of olive oil. Stir gently with a fork until combined. Set aside.
Combine flour and salt in a separate bowl. Add to the flour mixture your onions, and rosemary in alternating ingredients, stirring gently until dough is combined. (It’ll be sticky!)
Generously flour a flat surface. Knead dough 15 to 20 times, adding flour generously to make it easier to handle. (Keep it sticky, though!) Drizzle olive oil in a separate bowl and then add your ball of dough, turning to coat. Cover with a tea towel and place in a warm spot for 1 1/2 to 2 hours.
Preheat oven to 400 degrees. Divide dough into twelve portions, and form each into a rough round shape. Place on a baking mat or parchment paper, then cover with a towel and allow to rise for 15 to 20 minutes. Bake for 18 to 20 minutes or until brown. Brush with butter halfway through baking time.