Preheat a grill to medium-high heat. Season steaks with salt and pepper.
On low heat, melt butter in a heavy-bottomed sauce pan. Add flour slowly and stir to incorporate. Cook until it begins to foam and slightly darkens, about 3 minutes. Add milk and stir vigorously. Cook until mixture thickens and forms a smooth consistency, then add cheese. Remove from heat and add salt and pepper to taste.
Place steak on the grill and grill each side for about 4-5 minutes. Remove from grill and let rest for 5 minutes. Slice into 1 inch pieces and drizzle with béchamel sauce.