Mix the filling: finely chop the dried tomatoes and garlic (or pulse in a blender), combine with the dried herbs (thyme, oregano, rosemary), salt, and black pepper to form a paste.
Prep the pork: remove any tendons and fat from each pork fillet. Trim the thin tapered ends so both fillets are evenly thick. Using a long sharp knife, carefully butterfly each fillet lengthways into a flat piece about 1.5 cm thick.
Assemble the roulades: spread the tomato and herb filling evenly on both butterflied pieces. Roll each tightly into a cylinder and tie with kitchen twine at 2-3 cm intervals to secure.
Sear and slow-roast: heat 1 teaspoon of olive oil in a large oven-safe pan over medium heat. Sear the filled roulades for 2-3 minutes per side until lightly browned. Preheat the oven to 80°C (176°F) with no convection setting. Transfer the pan to the oven and cook the roulades for 1 to 1.5 hours, until the internal temperature reaches 63°C (145°F) when measured with a meat thermometer in the thickest part.
Prepare the sauce vegetables: while the roulades finish cooking, roughly dice the carrot and onion. In another pan, heat a little more olive oil and fry the diced vegetables over medium heat, stirring occasionally, until they begin to brown, about 5 minutes. Deglaze the pan with a little vegetable stock, scraping up the browned bits.
Build the sauce: transfer the seared vegetables and pan drippings to a saucepan. Pour in the remaining vegetable stock. Bring to a simmer and cook for about 1 hour to allow the flavors to infuse and the liquid to reduce. Pour the liquid through a fine sieve to strain out the vegetables. Return the strained sauce to the pan and continue simmering until it reduces to about 1/3 of its original volume. Remove from heat, season with balsamic vinegar and salt to taste, then whisk in the cold butter until emulsified.
Serve: remove the roulades from the oven, remove and discard the kitchen twine, and slice diagonally into even pieces. Arrange on a plate and spoon the finished sauce over and around the meat.
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