Trim the pork cheeks by cutting away any excess fat and connective tissue; season generously with salt and pepper.
Dredge the cheeks lightly in all-purpose flour and shake off any excess.
Heat extra-virgin olive oil in a Dutch oven or large heavy pot over medium-high heat. Add the cheeks and brown on all sides, about 4–5 minutes per side. Transfer to a plate and set aside.
Reduce heat to medium-low. Add onions, carrots, and garlic to the pot (add a little more oil if needed); cook, stirring occasionally, until the vegetables are soft and lightly golden, about 10–15 minutes.
Increase heat to medium-high, pour in the dry red wine, and bring to a simmer. Cook for about 5 minutes, scraping up any browned bits from the bottom of the pot.
Return the pork cheeks to the pot and add beef broth, bay leaves, and fresh thyme. The liquid should come about three-quarters of the way up the meat; add a splash of water if needed.
For stovetop cooking: cover and simmer over low heat until the cheeks are fork-tender, 2½–3 hours. For a pressure cooker: lock the lid, bring to high pressure, and cook for 45–60 minutes, then allow a natural release.
Remove the pork cheeks and set them aside. Discard the bay leaves and thyme stems.
Use an immersion blender to purée the cooking liquid and vegetables until smooth. Return the sauce to medium heat and simmer until slightly thickened, then adjust seasoning with salt and pepper.
Return the pork cheeks to the sauce, warm through gently, and serve immediately.