Trim off just enough excess skin from the chicken thighs to leave a sizable pouch to stuff. Season the chicken all over with salt.
In a medium mixing bowl, combine the broccolini, pickled chiles, almonds, garlic, lime zest, lime juice, cumin, black pepper, 2 tablespoons olive oil and 1 teaspoon salt. Mix well with a fork.
Preheat the air fryer to 350ºF with low convection. Working on a baking sheet, gently loosen and lift the skin from the thighs to make a pocket that’s still attached around most of the edges. Divide the filling into six equal portions and, using your fingers, fill each thigh pocket with the filling. Wrap the skin back over the filling and meat. Drizzle the chicken with the remaining 2 tablespoons oil.
If you have a large format air fryer, place the chicken thighs so they fit snugly in an 8-by-8-inch square baking dish. Place the dish in the air fryer and cook until the skin is crisp and golden and the meat is cooked through (an instant-read thermometer stuck into the thickest part away from the bone should read 165ºF), 25 to 30 minutes. If you have a regular-sized air fryer with a drawer, place the thighs in the drawer and cook the same way. Serve hot.