Preheat the oven to 400°F (200°C). Lightly brush a 9x13-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 half of the fajita seasoning; toss to coat.
Cook the chicken about 5 minutes, stirring occasionally, until no longer pink and lightly charred. Remove the chicken to a plate and wipe the skillet clean.
Add the remaining 2 tablespoons of butter to the skillet and melt. Add the sliced onions and bell peppers, season with the remaining fajita seasoning, and sauté a few minutes until softened and onions are translucent.
Return the cooked chicken to the skillet and toss with the vegetables until combined and heated through.
Lay a tortilla flat, add a portion of the chicken and pepper mixture and about 2 tablespoons of shredded cheese, then tightly roll the tortilla and place seam-side down in the prepared baking dish. Repeat with remaining tortillas and filling.
Brush the tops of the rolled taquitos with the melted butter.
Bake in the preheated oven about 15 minutes, until tortillas are golden brown. If needed, broil 1–2 minutes to deepen the color, watching carefully to avoid burning.
Remove from the oven and serve warm with salsa and/or sour cream if desired.
