Pat the pork dry and season with salt and pepper. Heat oil in a Dutch oven over medium-high heat and sear the pork in batches until deeply browned on at least two sides, about 3–4 minutes per batch. Remove and set aside.
In the same pot, sauté the diced onion for 4–5 minutes until soft, then add the garlic and cook for 1 more minute.
Return the pork to the pot along with the roasted green chiles, diced tomatoes, chicken broth, cumin, oregano, and paprika. Bring to a simmer, partially cover, and cook for 60–75 minutes, until the pork is fork-tender.
While the chili simmers, melt the butter in a small saucepan over medium heat and whisk in the flour. Cook for 2 minutes, whisking constantly, until lightly toasted-smelling.
Whisk 1 cup of the hot chili liquid into the roux until smooth, then stir the tempered roux back into the pot.
Simmer uncovered for 15 more minutes, until the chili coats the back of a spoon and holds a clean line when you drag a finger through it. Taste and adjust salt.
Serve hot with cilantro, diced onion, lime wedges, and warm flour tortillas.