Pat the pork cubes dry and toss them with the flour, salt, and pepper until evenly coated.
Heat the olive oil in a Dutch oven over medium-high heat. Add the pork in batches and brown on all sides, then set the meat aside.
Lower the heat to medium and add the onion and carrots to the same pot. Cook until the onion softens, about 5 minutes.
Stir in the garlic and tomato paste and cook for 1 minute until fragrant.
Pour in the red wine and scrape the bottom of the pot to release any browned bits. Let the wine simmer for 2 minutes.
Return the pork to the pot along with the beef broth and thyme. Bring to a gentle boil, then reduce to low heat, cover, and simmer for 1 hour.
Add the sliced mushrooms, cover again, and continue simmering for 30 more minutes until the pork is fork-tender and the sauce has thickened slightly.