How to Make It
Heat oil in a medium skillet over medium-high. Add 1/2 cup of the onion; sprinkle with salt and pepper, and cook, stirring often, until translucent, 3 minutes. Stir in beans, chiles, water, and cumin; cook, stirring occasionally, until mixture has slightly thickened and warmed through, about 3 minutes. Stir in chicken and 2 teaspoons of the lime juice; cook 2 minutes. Remove from heat.
Stir together tomatoes, jalapeño, remaining onion and lime juice, and if desired, cilantro in a small bowl; set aside.
Place 2 tablespoons of the cheese along the center of each tortilla. Divide chicken filling over cheese, and top with 2 more tablespoons cheese. Tuck sides of tortillas partially over filling, and roll to form a burrito. Brush burritos with melted butter.
Preheat air-fryer to 400°F for 2 minutes. Place 2 burritos in at a time, and cook 8 minutes, flipping halfway through. Repeat with remaining 2 burritos. Top each with 2 tablespoons of the queso dip, and about 1 tablespoon of the tomato mixture.