Melt the butter in a small saucepan over medium-low heat.

Whisk in the flour and cook for 1 minute, stirring constantly, until it smells faintly toasted and loses its raw edge.

Pour in the milk gradually, whisking the whole time, until the mixture is smooth with no lumps.

Cook for 3 to 4 minutes, stirring often, until the sauce thickens enough to coat the back of a spoon.

Reduce the heat to low and add the American cheese and pepper jack a handful at a time, stirring until each addition fully melts before adding the next.

Stir in the green chiles.

Pour into a warm bowl right before serving, and set it into the board with the tortilla chips.