1. Preheat oven to 450ºF. Place tomatoes on a foil-lined baking sheet, cut side up. Drizzle with 2 Tbsp. olive oil and sprinkle with salt and pepper. Bake for 25 to 30 minutes, until soft and wrinkled but not burned. Let cool for 5 minutes.
2. In a large heavy skillet over medium heat, warm remaining 1 Tbsp. oil and cook onion and garlic for 2 to 3 minutes, stirring often. Put tomatoes and onion mixture in a food processor and process until pureed, about 30 seconds. Transfer mixture back to skillet, add tomato paste and brown sugar and cook over medium-low heat, stirring, for 10 minutes. Stir in both vinegars and pickling spice, reduce heat to low and simmer until very thick, stirring occasionally, about 1 hour 30 minutes.
3. Strain mixture through a fine-mesh sieve, pressing on solids to force all liquid through. Discard solids. Season mixture with salt and pepper and let cool before serving (it will thicken as it cools).