Preheat the oven to 375 degrees F. Line a baking sheet with parchment paper.
Place the tomato halves flesh-side up in a single layer on the prepared baking sheet. In a medium bowl, whisk together 4 tablespoons of the olive oil, the confectioners' sugar, red pepper flakes and 2 teaspoons salt. Spread the mixture evenly over the tomato halves. Roast the tomatoes until tender, 35 to 40 minutes.
Heat a large skillet over high heat and add half of the canola oil. When it begins to smoke lightly, add half of the poblanos and cook to tenderize without making them mushy. Using a slotted spoon, remove the peppers to a baking sheet, cavity-side up. Repeat with the remaining poblanos and canola oil. Season with salt and set aside.
In a large skillet, heat the 3 tablespoons of the olive oil and the thyme. As the oil heats, the thyme will fry and become crispy. Use a slotted spoon to remove the thyme sprigs, then season with salt and cool. Add the bell peppers to the same skillet and season with salt. Cook over high heat, stirring with a wooden spoon, until the peppers brown slightly but are still slightly firm, 3 to 5 minutes. Use a slotted spoon to remove the peppers and transfer to a medium bowl. Add the remaining 1 tablespoon olive oil to the skillet, heat until it begins to smoke lightly, and add the garlic and onions. Reduce the heat to medium. Season with salt and cook until the onions are lightly brown and tender, 8 to 10 minutes. Add the onions to the bell peppers.
Quarter the oven-dried tomatoes and add to the pepper and onion mix. Chop the basil leaves and stir into the filling along with the red wine vinegar and mozzarella. Pull the thyme leaves off the stems and add to the mixture. Taste for seasoning and adjust if necessary.
In a medium bowl, stir together the breadcrumbs, Parmesan and melted butter. Stuff each pepper halfway with the filling, and then top with the breadcrumb mixture.
Heat the peppers in the oven to brown the tops, 10 to 12 minutes. Arrange on a platter and drizzle with the balsamic vinegar to serve.
Remove all of the seeds and inner ribs from the poblanos so they lose their heat.