Preheat oven to 350 degrees.
Heat the olive oil over medium-high heat in a Dutch oven large enough to hold the short ribs in one layer.
Add the bacon and cook for 5-6 minutes, frequently stirring until the bacon is browned and crisp, adjusting the heat as necessary so the bacon doesn't burn. Transfer to a paper towel-lined plate. Reserve the drippings in the Dutch oven and keep the heat at medium-high.
Season the short ribs generously with salt and black pepper. Place the ribs in the Dutch oven and brown for 3-4 minutes on each side or until nicely browned, adjusting the heat as necessary so the fat does not smoke.
Add the onions, reduce heat to medium-low and cook for 3-4 minutes or until beginning to soften. Add the garlic and cook 10-15 seconds or until fragrant.
Add the tomato paste and stir.
Add the wine, bring to a boil and cook 2-3 minutes to reduce slightly.
Add the broth, bay leaves and thyme. Bring to a boil.
Transfer the short ribs back to the Dutch oven along with the bacon. Cover the top of the Dutch oven securely with a piece of parchment paper or aluminum foil then place the lid on the Dutch oven.
Place in the oven and cook for 2 hours.
Add the carrots and if using fresh pearl onions, then add them. (If using frozen, thawed pearl onions, add them later when adding the flour.) Return the Dutch oven to the oven and cook another 30-45 minutes or until the beef, carrots and pearl onions are tender.
If using frozen, thawed pearl onions, add them at this point.
Transfer to your stovetop.
Combine flour and 2 tablespoons butter in a small bowl. Remove 3-4 tablespoons of the hot braising liquid from the Dutch oven and add it to the butter and flour. Stir until you have a thick paste slurry.
Add the slurry (flour, butter and liquid mixture) to the Dutch oven and gently stir it in.
Bring to a gentle simmer on your cooktop and cook for 4-5 minutes or until thickened. Cover to keep warm.
Heat the remaining butter in a small nonstick skillet or saute pan. Add the mushrooms and cook over high heat or until browned and tender and they release their juices. Add to Dutch oven and stir them in.
Add the fresh parsley. Taste and adjust the seasoning with salt and black pepper.