Melt the stick of butter in a large skillet over medium-low heat. Once melted, add in the flour and whisk to incorporate. Continue to whisk for about 4-5 minutes, cooking out the raw flavor of the flour, and make a golden, smooth roux.
Slowly pour the warmed dairy into the roux using 1/2 cup at a time, whisking to thicken the sauce. Add in the spices and whisk constantly while the sauce is simmering, just until thickened.
Once the sauce has just started to thicken, turn off the heat. Slowly add in the cheese about 1/2 cup at a time and whisk to melt.