Trim fat from the pork roast. Cut the pork into 2-3 large pieces and season each piece with salt and pepper on all sides.
Heat oil in a large skillet over high heat. Add all the pieces of pork and sear for 1 minute on each side, just until browned. Transfer the pork to slow cooker.
Add the liquid smoke and 1/2 cup water. Cook on high for 4-5 hours or on low for 8 hours.
For the Rice:
While the pork is cooking, prepare the rice. Melt butter in a saucepan over medium-high heat. Add the rice.
Cook, stirring, for 2 minutes. Add chicken broth, salt, pepper, lime juice and zest, cilantro and cumin and bring to a boil.
Reduce heat, cover and cook for 18-20 minutes. Remove from heat and let stand, covered, for 10 minutes. This can be made ahead and refrigerated until ready to assemble the burritos.
Remove the pork from the slow cooker and shred into bite-sized pieces (discard any pieces of fat). In a large bowl, toss the shredded pork with the brown sugar, 1/2 cup of the green sauce, chili powder, salt and pepper.
Add the beans, rice and cilantro to the pork mixture.
Lightly grease a 9X13-inch baking pan and spoon some of the green sauce on the bottom.
Scoop about 1/2 cup of the meat mixture into each tortilla and sprinkle with a little bit of cheese Roll up the tortillas and place seam-side down in the baking dish. Pour the remaining sauce over them and sprinkle with remaining cheese.
Bake at 350 degrees for 15-20 minutes until heated through and the top is bubbly and slightly golden. Serve immediately.