Step 1

In a bowl, combine all of the ingredients and season with salt.

Step 2

In a saucepan, melt the butter. Stir in the flour and cook over moderate heat for 30 seconds. Whisk in the milk and cook, whisking, until thickened, 5 minutes. Stir in the Jack cheese until melted, then stir in the pecorino; season with salt and pepper. Let cool until spreadable.

Step 3

Light a grill. Form the beef into 4 patties and brush with oil; season with salt and pepper. Grill over moderately high heat until browned outside and medium within, 4 minutes per side. Place the burgers on the buns, top with the cheese sauce, salsa, jalapeños and chips, and serve.