Heat 1 tablespoon vegetable oil in a large heavy-bottomed saucepan or Dutch oven over medium-high heat.
Add the chicken thighs and sear undisturbed 3–4 minutes per side until browned; transfer to a bowl and set aside.
Return the pan to medium-high heat and add the chopped onion, poblano, and jalapeño; cook about 5 minutes until softened.
Add the chopped tomatoes and cook, stirring occasionally, about 5 more minutes.
Return the browned chicken and any accumulated juices to the pan, arrange in a single layer over the vegetables, add 1 cup chicken stock, bring to a simmer, then reduce heat to low and partially cover.
Simmer gently, partially covered, 45–60 minutes until the chicken is very tender and easily shreddable.
Meanwhile, cook the basmati rice according to package instructions; when done, fluff and stir in chopped cilantro, 1 tablespoon lime juice, and salt and pepper to taste.
Remove the chicken to a cutting board and shred with two forks when cool enough to handle.
Continue simmering the vegetables and remaining liquid in the pan uncovered over medium heat until thickened, about 10 minutes, then return shredded chicken and stir to coat; remove from heat.
Warm the refried beans in a small saucepan and thin with a little water or broth if needed; season to taste.
Preheat the oven to 300°F and warm a nonstick skillet or flat griddle over medium heat.
Warm the tortillas in the preheating oven wrapped in a damp towel so they are soft and pliable.
To assemble each burrito: spread a few spoonfuls of refried beans on the bottom third of a tortilla, add about 1/4 cup rice, then spoon on shredded chicken and top with shredded cheese.
Fold in the sides slightly and roll tightly from the bottom to enclose the filling.
Place two assembled burritos seam-side down on the hot skillet, weigh them lightly with a medium flat pot if available, and cook 3–5 minutes until the bottom is golden; flip and cook 3–4 minutes more until both sides are browned and crispy.
Transfer cooked burritos to a baking sheet and keep warm in the oven while you crisp the remaining burritos.
Serve the burritos warm with red or green salsa and optional sour cream.
