Heat the oil in a large saute pan over medium-high heat. Add the onion and celery and saute for 1 minute just to soften. Add the garlic and season with salt and pepper cooking for about 3 minutes. Add the tomato paste and thyme and continue cooking for another 2 minutes. Deglaze with the beef broth using a wooden spoon to pull up any bits from the bottom of the pan. Season with salt and pepper.
Carefully transfer mixture to the slow cooker. Stir in the canned tomatoes. Mix the pork and sirloin together in a separate bowl, using your hands, until combined. Stir the meat into the slow cooker, spreading out evenly and avoiding any large clumps. Cover and cook on high for 4 to 6 hours or on low 8 to 10 hours. During last two hours of cooking, add canned carrots and mushrooms to ragu. Skim the accumulated grease from the top before serving.
Serve over pasta with crusty bread for soaking up the sauce.