Preheat the oven to 400°F (200°C). Lightly brush a 9×13-inch baking dish with some of the melted butter.
Heat a large skillet over medium-high heat and melt 1 tablespoon of the butter. Add the chicken strips and season with half (1 tablespoon) of the fajita seasoning. Cook about 5 minutes until no longer pink and slightly charred; remove the chicken and set aside.
Wipe the skillet clean, then add the remaining 2 tablespoons of butter and melt. Add the sliced onions and bell peppers, season with the remaining fajita seasoning, and cook 2–4 minutes until the vegetables soften and the onions are translucent.
Combine the cooked chicken with the sautéed onions and peppers in the skillet and stir to mix evenly.
Lay a tortilla flat, add a portion of the chicken and vegetable mixture, and sprinkle about 1–2 tablespoons of shredded Mexican cheese on top. Tightly roll the tortilla and place it seam-side down in the prepared baking dish. Repeat with remaining tortillas and filling.
Brush the rolled taquitos with the remaining melted butter, then bake in the preheated oven for about 15 minutes, or until the tortillas are golden brown.
If the taquitos are not golden enough after baking, broil 1–2 minutes until desired color, watching carefully to prevent burning.
Serve warm with salsa and/or sour cream if desired.