I start by heating oil in a large saucepan over medium-high heat, then adding the chicken thighs in a single layer. I let them sit undisturbed for a few minutes so they actually brown instead of steam. Once both sides are golden, I pull the chicken out and set it aside while I build the sauce.
In the same pan, I add the chopped onion, poblano, and jalapeño, letting them soften over medium heat for about five minutes. Then I stir in the chopped tomatoes and let everything cook a few minutes more. This step builds most of the flavor base for the whole filling.
I return the browned chicken to the pan along with any juices, then pour in the chicken stock. Once it comes to a simmer, I partially cover the pan and let it cook low and slow for about 45 to 60 minutes, until the chicken practically falls apart on its own.
Once the chicken is tender, I move it to a cutting board and shred it with two forks while the sauce keeps simmering uncovered. This reduces the liquid into a thicker, saucier base. I stir the shredded chicken back into the pan so every piece gets coated.
While the chicken simmers, I cook the basmati rice according to the package, then fold in chopped cilantro and a splash of lime juice once it's done. I warm the refried beans separately in a small saucepan, thinning them slightly with water if they feel too thick.
I preheat my oven to 300 degrees and warm the tortillas inside, covered with a damp towel, so they steam soft and pliable. Then I spread beans, rice, and chicken across the bottom third of each tortilla, leaving space on the sides, and top it all with a generous layer of shredded cheese.
I fold in the sides, then roll each tortilla tightly from the bottom up, tucking as I go. I place two burritos seam-side down on a hot skillet, set a flat-bottomed pot on top to weigh them down, and toast for a few minutes per side until deeply golden and crisp.