In a shallow bowl, beat the egg. In a second shallow bowl, combine the crushed crackers and cheese.
Dip each chicken strip in the egg, allowing extras to drip back into the bowl, and then coat it with the cracker and cheese mixture. Set them aside on a plate.
In a large skillet, melt 1 Tbsp. of butter with 1 Tbsp. of oil over medium heat.
Cook the tenderloins in one layer for about 4 minutes per side, until they are nicely browned. Continue until you have cooked all the chicken, adding the additional butter and oil to the pan if needed.
Transfer the cooked tenders to a clean plate. If you are using the skewers, push them lengthwise through some or all of the chicken. (If the skewers are sharp, push them all the way through so the kids won’t accidentally bite the ends.) Serve the chicken sticks immediately with one or more of the dipping sauces.