In a sauce pan over med-high heat, melt the butter. Add the flour & cook, stirring, until it forms a blonde roux. Add in the milk & bring to a simmer. Stir often & continue to cook until thickened. Stir in the cheese until melted & well blended. Adjust seasoning with salt & pepper.
Heat a nonstick sauté pan over medium to med-high heat. Add a little olive oil & crack in an egg (being careful not to break the yolk). Season with salt & pepper. Pour about 1 tsp of water into the edge of the pan & cover. Cook for about 3-4 minutes until the yolk is at desired doneness. Repeat with the remaining eggs.
Build open face sandwiches with the toasts topped with ham, then egg, sauce & garnished with parsley.
