In a large saucepan, melt butter. Stir in flour until smooth; gradually add milk.
Bring to a boil; cook and stir for 2 minutes or until thickened.
Reduce heat; stir in the cheeses and seasonings. Cook and stir until cheese is melted; keep warm.
Place 2-3 in. of water in a large skillet with high sides; add vinegar.
Bring to a boil; reduce heat and simmer gently.
Break cold eggs, one at a time, into a custard cup or saucer; holding the cup close to the surface of the water, slip each egg into water.
Cook, uncovered, until whites are completely set and yolks begin to thicken (but are not hard), about 4 minutes.
Place Canadian bacon and avocado on each muffin half.
With a slotted spoon, lift each egg out of the water and place over avocado.
Top with cheese sauce.