Heat oil in a large pot over medium-high heat; brown pork seasoned with salt and pepper.
Stir in roasted green chiles, tomatillos, onion, garlic, cumin, and oregano; cook 2 minutes.
Add chicken broth; bring to a boil, reduce heat, and simmer until pork is tender, about 60 minutes.
Meanwhile, wrap tortillas in foil and warm in a 350 F oven until pliable, about 10 minutes.
Assemble burritos: divide pork mixture and rice among tortillas, top with cheese and cilantro, and roll up.