Melt butter in a large pan over medium heat. Once melted, add the chopped leeks, salt, and pepper, and cook until leeks are soft and fragrant, about 3-4 minutes.
Add the flour, stirring to coat the leeks, and cook for 1 minute. Add the ale, milk, mustard, and cayenne. Bring to a boil, then reduce heat to low and stir until sauce is thick enough to coat the back of a spoon, about 2-3 minutes.
Stir in the grated cheddar and cider vinegar, mixing constantly until smooth. Mix in the egg yolk, again stirring constantly until sauce is completely smooth. Season to taste with additional salt if needed.
Preheat broiler, and arrange toast slices on a foil-lined baking sheet. Top each piece of toast with an equal amount of the cheese sauce. Place under broiler, and broil until cheese sauce is bubbling and starting to brown (keep a close eye on this to prevent burning). Top with fresh chives and serve immediately, ideally with a cold beer.