Position an oven rack in the middle position. Preheat the oven to 325ºF.
Place the pork (fat side up) in a large dutch oven. Season with the salt and pepper. Scatter the thyme, garlic, bay leaves, and onion around the meat. Add the wine and bring to a boil. Cover the meat with a round of parchment paper and then with a tight fitting lid. Place the pot in the oven to braise until the meat is very tender and falling off the bone, about 4-5 hours.
Remove the pot from the oven and cool, partially covered, for 30 minutes. Transfer the pork to a platter, tent with a sheet of foil, and cool completely. Remove the thyme and bay leaves from the braising liquid. Strain the braising liquid into a bowl; reserve the onions. Skim off any fat from the braising liquid. If there are more than 2 cups of liquid, simmer gently in a pot until reduced. If less than 2 cups, add chicken broth.
Once cool, trim any excess fat from the pork and remove the meat from the bone. Shred the meat into 1 ½ inch pieces and return to the pot with the onions. Bring the braising liquid to a simmer in a small saucepan.
In a large skillet, heat the butter over medium heat until foaming. Stir in the flour and cook over medium-high heat, stirring constantly, until light to medium brown in color, about 2 minutes. Slowly whisk the braising liquid into the roux, whisking constantly, and cook until thickened. Season to taste with salt and ground pepper. Spoon about 1 cup of the sauce over the meat and onions and cook over medium-low heat until warmed through. Serve the remaining sauce on the side. Braised pork can be made 2 days ahead. Chill the meat and onions with 1 cup of sauce, and chill the remaining sauce separately. Enjoy!