Make basil balsamic dressing. Combine balsamic vinegar, basil, honey, dijon mustard, and black pepper in a blender. Slowly add oil while the blender is running and blend until smooth. Set aside.
Make red bell pepper aioli. Blend together mayo, red bell pepper, lemon juice, and garlic until smooth. Season with salt and pepper and set aside.
Prepare fried burrata. Pat burrata dry. Combine flour, ½ teaspoon salt, and ¼ teaspoon black pepper in a shallow bowl. Beat egg in another bowl and place panko in a third bowl. Dredge burrata in flour mixture, coat in egg, and coat in panko. Dip in egg and coat in panko again. Place burrata on a plate and chill in the freezer for 20 minutes.
Heat 2 tablespoons oil in a large saute pan over medium heat. Lay eggplant slices in pan and cook until browned, about 4 minutes. Flip eggplant over and season with oregano, salt, and pepper. Continue to cook until browned and tender, another 3-4 minutes. Remove from heat and keep warm.
Heat frying oil to 375 degrees F.
Carefully drop burrata in oil and deep fry until golden brown, about 1 minute. Flip over and fry until golden brown on the other side, about 1 minute. Remove burrata with a slotted spoon and drain excess oil on paper towels.
Assemble sandwiches. Spread bell pepper aioli on half of the sourdough slices. Top with mesclun greens, sliced tomatoes, burrata, and roasted eggplant. Drizzle basil balsamic dressing on top and stack with remaining bread. Serve immediately.