On a gas or charcoal grill over medium-high heat, grill onions, and all peppers until blackened and onions are tender and browned. Remove them from the grill, and place them in foil, or plastic Ziplock bags until they are cool enough to handle. When cool, remove the stems and seeds and blackened skin from the peppers and discard. Chop the peppers and onions and set aside.
Place cut and seeded tomatoes into a 6- or 7-quart Dutch Oven. Stir in brown sugar, vinegar, salt and paprika to the pot. Bring the mixture to a boil, allowing to cook down for about 50 minutes, stirring frequently. Stir in the grilled onions, grilled peppers, and garlic. Continue to cook for 30-40 minutes, until it reaches desired consistency. To break down the sauce, blend the sauce to your desired consistency with an immersion blender. Leave it with some chunks of grilled vegetables. Remove and stir in the rosemary.
To process for long term storage, pour one tablespoon of lemon juice into the bottom of hot canning jars. Ladle the hot pasta sauce into the jars on top of the lemon juice, leaving ½” headspace in the jars. Wipe the rims of the jars clean, and place lids and bands on sealing to finger tight.
Process the jars in a boiling water canner for 35 minutes, starting the timer when the water reaches boiling. Remove from the canner and cool on a wire rack to seal and cool completely.