Place one oven rack in the top position. Place another oven rack in the middle position. Heat the oven to 350 degrees.
Place the bread on a sheet pan. Brush the melted butter onto the slices of bread. Place into the oven and toast at 350 degrees for 4-6 minutes. Remove from the oven. Turn off the oven and turn on the broiler to it's highest setting.
In a saucepan melt the 2 tablespoons of butter. Whisk in the flour and let cook for 2 minutes. Add the dry mustard and salt. Whisk to combine.
Add the stout and Worcestershire to the mixture and continue to whisk until combined. Add the cheese and let melt, stirring occasionally about 2-3 minutes.
Remove from the heat and while whisking constantly, add the egg yolk one at a time. Continue to whisk until all ingredients are incorporated. Spread the sauce onto the toasted bread. Place back into the oven and broil for 1-2 minutes, or until bubbly and browned. Serve warm.
I prefer Welsh Rarebit to have a mustardy, sharp bite. If you prefer a more Stout-forward bite, decrease the amount of mustard to 1 1/2 teaspoons and the white cheddar to 1/2 cup.