Melt 4 tablespoons of the butter in a medium skillet over medium heat. Stir in the flour and cook for 1 to 2 minutes. Pour in the cream and bring to a simmer. Turn off the heat and add the salt and cheeses. Let sit for a minute or 2, and then stir to melt the cheeses completely. Cover and keep warm.

Bring a medium skillet of water to a very gentle simmer and add the white vinegar. Crack an egg into a small dish and carefully slip it into the water. Cook for 2 to 3 minutes, depending on how hard or soft you like your yolk. Remove with a slotted spoon onto a paper-towel-lined plate. Continue with the rest of the eggs.

Split the baguette and toast it under the broiler. Slather with the remaining butter and cut into slices.

Fill individual bowls with the cheese mixture and top with the poached eggs. Garnish with chives and serve with the toasted bread.