Remove the stem end of the jalapeno and slice lengthwise into quarters.
Melt butter in a saucepan on medium-low heat and add jalapeno quarters, salt, and pepper. Stir the jalapeno and seasonings in the butter as it melts to infuse it with the flavor of the pepper and spices.
Sprinkle in flour and stir to make a roux. Cook the flour mixture for 2-3 minutes while continuing to stir, to cook out the raw flour flavor.
Stir in half and half by adding a few splashes at a time, stirring to fully incorporate before more liquid is added. Repeat with milk as well.
Heat sauce mixture to almost boiling, then turn heat to low and beginning adding Monterey Jack cheese a handful at a time, stirring to melt into the sauce before more cheese is added. Once Monterey Jack cheese has been added, repeat the process with Cheddar cheese.
Once cheese has been added to the sauce, check for proper thickness. Add a tablespoon of milk at a time if you need to thin the sauce out. If the sauce gets too thin, you can heat it longer to thicken up. Jalapeno peppers can be removed for serving.