Heat olive oil in a 4-quart Dutch oven over medium heat. Sauté the onions with the carrot and salt until just starting to turn golden. Add garlic cloves and cook for another minute. (About 10 minutes total.)
Stir in the tomato paste.
Add the shredded brisket. Then add wine, bay leaf and thyme. Stir together.
Bring to a simmer and cook for 30 minutes or until the desired sauce consistency is reached, stirring every 5 to 10 minutes to redistribute the flavors.
Be sure to taste before adding any salt or pepper, since your brisket might have plenty of seasoning. And pasta should be cooked with plenty of salt in the water (2 tablespoon per pound of uncooked pasta).
Serve mixed into pasta or gnocchi (or serve over polenta or risotto) with a sprinkle of grated Parmesan and chopped parsley.
Optional: Add crushed tomatoes or tomato sauce after the tomato paste if you want a more tomato-y texture.