Whisk together olive oil, lime juice, minced garlic, cumin, smoked paprika, and salt in a shallow dish. Add chicken and turn to coat. Marinate at room temperature while you prep the rice, or refrigerate up to 4 hours.

Rinse rice until water runs clear. Combine with chicken broth and 1/2 teaspoon salt in a saucepan. Bring to a boil, then reduce heat to low, cover, and simmer for 15 minutes until liquid is absorbed. Remove from heat, let stand 5 minutes, then fluff with a fork. Stir in half the cilantro and all the lime zest.

Heat grill or grill pan to medium-high. Remove chicken from marinade, letting excess drip off. Grill 5-6 minutes per side until charred in spots and internal temperature reaches 165 degrees F. Transfer to a cutting board and rest 5 minutes, then slice against the grain.

Working one at a time, warm tortillas directly over a gas flame or in a dry skillet for 10-15 seconds per side until pliable with light char marks. Keep wrapped in a clean towel.

Lay a tortilla on your work surface. Mound 3/4 cup rice slightly off-center, top with a quarter of the chicken, 1/4 cup cheese, a few avocado slices, and a dollop of sour cream. Fold sides in, then roll tightly from the bottom, tucking as you go. Place seam-side down. Repeat with remaining burritos.

For a crispy exterior, place assembled burritos seam-side down in a hot dry skillet and cook 1-2 minutes per side until golden. Serve with remaining cilantro scattered on top.