Preheat oven to 350 degrees.
Add the onion, carrot, celery, and garlic to food processor and pulse several times until the vegetables are all minced uniformly, but not mushy.
Season the lamb shanks all over with Salt and Pepper. In a large Dutch oven over medium-high heat, heat 2 tablespoons of the olive oil until almost smoking then add the lamb shanks to sear on all sides. Once seared all over, remove and set aside.
In the same pot add the remaining olive oil and then the minced vegetables stirring well to coat with the oil. Cook until vegetables are transparent about 4-5 minutes. Add the tomato paste and incorporate into the vegetables. Let the mixture “toast” the tomato paste mixture slightly. Add the dried thyme and mix well. Add the white wine and deglaze the pot and let the mixture reduce for approximately 3 minutes until the mixture is a nice creamy base. Salt and Pepper to taste.
Add the red wine and beef stock to the mixture along with the bay leaves and stir well to incorporate all of the ingredients. Add the lamb shanks back to the pot and bring the mixture back to a simmer. Cover the pot and place into the preheated oven for 45 minutes. After the first 45 minutes turn the shanks over and continue to braise, covered, for another 45 minutes. Remove from the oven and let cool slightly.
Once the meat has cooled enough to work with it by hand, remove the shanks to a large bowl. Start by removing the outer fat layer and separate the “lobes” of meat from the bone discarding the fat and connective tissue. Cut larger pieces of meat into smaller sizes to be mixed with the vegetables later. Reserve the braising liquid.