Heat 1 tablespoon oil in a medium saute pan over medium-high heat. Add chopped onion and saute until browned, about 10 minutes, stirring frequently. Remove from heat and let cool completely.
Combine warm water with sugar and active dry yeast in a medium bowl. Let sit for 10 minutes or until mixture is foamy.
Add flour, salt, yeast mixture, and remaining 2 tablespoons olive oil in the bowl of a stand mixer. Knead dough for 5 minutes. Add ½ cup cooked onion (100 grams), Asiago cheese, and rosemary to the dough and continue to knead for another 3-4 minutes or until well incorporated. Add 1-2 tablespoons extra flour if dough is too wet.
Transfer dough to a clean bowl and cover. Let rise in a warm area for 1 hour or until doubled in size.
Preheat oven to 450 degrees F.
Punch down dough and divide into 5 equal portions. Roll each portion into a ball and let sit for 10 minutes.
Mix together ½ cup water with molasses in a wide bowl. Add sesame seeds in another wide bowl. Line baking sheet with parchment paper.
Lightly flour a clean work counter. Working with one portion at a time, roll dough into 30-inch long rope. Fold rope in half. Grab each end and twist the rope in opposite directions. Pinch the two ends together to seal. Repeat with remaining portions.
Dip dough in molasses mixture, coating both sides. Dredge dough in sesame seeds, turning to coat. Place dough on prepared baking sheet and repeat with remaining portions. Let sit for 30 minutes.
Spray top of bagels with water and bake for 10 minutes. Reduce oven to 400 degrees and rotate sheet pan. Continue baking for 8-10 minutes or until the tops and bottom are both golden brown. Remove from oven and let cool slightly. Serve warm or at room temperature.