01 - Rinse rice under cold water until water runs clear. Combine rice, water, and salt in a saucepan. Bring to a boil over high heat, reduce heat to low, cover tightly, and simmer for 15 minutes. Remove from heat and let stand covered for 5 minutes. Fluff with a fork, then stir in lime juice and chopped cilantro.
02 - Heat olive oil in a large skillet over medium heat. Add chopped onion and cook until softened, about 3 minutes. Add minced garlic and cook for 30 seconds until fragrant. Add ground beef, breaking apart with a spoon, and cook until browned. Drain excess fat if necessary. Stir in tomato paste, cumin, chili powder, smoked paprika, oregano, salt, and black pepper. Cook for 1 minute to bloom spices. Add beef broth and simmer for 5 minutes until sauce slightly thickens.
03 - Combine drained black beans, cumin, chili powder, salt, and pepper in a small saucepan over medium heat. Cook for 5 minutes, stirring occasionally, until heated through and well combined.
04 - Heat each tortilla in a dry skillet over medium heat for 30 seconds per side until warm and pliable, or microwave wrapped in damp paper towels for 30 seconds.
05 - Place a warm tortilla on a flat surface. Spoon portions of rice down the center, followed by beef mixture and beans. Layer with cheese, salsa, shredded lettuce, diced tomato, and sour cream. Fold in the sides, then roll from bottom to top, tucking ingredients in tightly. Repeat with remaining tortillas.
06 - Serve burritos immediately while warm. Accompany with additional salsa and sour cream on the side if desired.