Step 1: Make the Rosemary Pesto

1. In a food processor, combine rosemary leaves, basil leaves, garlic, pine nuts, and Parmesan cheese.
2. Pulse until coarsely chopped.
3. While the processor is running, slowly add olive oil until the mixture becomes a smooth paste.
4. Season with salt and pepper. Set the pesto aside.

Step 2: Prepare the Pesto Crust

1. In a large bowl, mix flour, salt, and grated Parmesan cheese.
2. Add the cold, cubed butter and use a pastry cutter or your fingers to blend until the mixture resembles coarse crumbs.
3. Add 2 tablespoons of the prepared rosemary pesto and mix until combined.
4. Slowly add ice water, one tablespoon at a time, until the dough forms a ball.
5. Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.

Step 3: Caramelize the Onions

1. Heat olive oil and butter in a large skillet over medium heat.
2. Add sliced onions and sprinkle with salt.
3. Cook, stirring occasionally, until the onions become soft and start to caramelize (about 20-25 minutes).
4. Add sugar (if using) and continue to cook for another 10 minutes for deeper caramelization.
5. Stir in balsamic vinegar and cook for an additional 2-3 minutes.
6. Remove from heat and set aside.

Step 4: Roll Out the Dough and Pre-bake

1. Preheat the oven to 375°F (190°C).
2. Roll out the chilled dough on a floured surface to fit a 9-inch tart pan.
3. Transfer the dough to the tart pan, pressing it into the edges and trimming any excess.
4. Prick the bottom of the crust with a fork and place parchment paper on top. Fill with pie weights or dried beans.
5. Bake for 15 minutes, then remove the weights and bake for an additional 5 minutes, until lightly golden.

Step 5: Assemble and Bake the Tart

1. Spread a thin layer of the rosemary pesto over the pre-baked tart crust.
2. Evenly distribute the caramelized onions over the pesto.
3. Tear the burrata into pieces and place it evenly over the onions.
4. Sprinkle grated Parmesan cheese on top.
5. Bake the tart for 15-20 minutes, or until the burrata is melty and slightly browned.

Step 6: Garnish and Serve

1. Remove the tart from the oven and let it cool slightly.
2. Garnish with fresh basil leaves and cracked black pepper.
3. Slice and serve warm.