In a medium saucepan over medium heat, make a roux by melting the butter and stirring in the flour. Add the beer or milk, mustard and Worcestershire sauce. Add 3/4 of the cheddar. Stir until combined.
Line a baking sheet with aluminum foil or parchment paper. Spoon Welsh Rarebit onto slices of a crusty bread on baking sheet. Top with remaining cheddar. Transfer to oven for 5-7 minutes, until cheese is melted. Serve warm.