Preheat the oven to 350 degrees F. Grease a 3-quart baking dish with 2 tablespoons of the butter.
Combine the milk, water, salt and the remaining 4 tablespoons butter in a medium saucepot; bring to a boil. Add the cornmeal in a steady stream while whisking rapidly. Continue stirring until the mixture returns to a simmer. Reduce the heat and simmer, stirring occasionally, until the mixture is very thick, 6 to 10 minutes. Remove the pot from the heat, stir in the cheese and add hot sauce to taste.
Crack the eggs into a small bowl and beat with a whisk. Add 1 cup of the hot cornmeal mixture to the eggs and whisk together to warm the eggs and prevent them from curdling when added to the pot. Continue adding spoonfuls of the hot cornmeal mixture to the bowl of eggs until it's warm to the touch; then whisk the egg mixture back into the pot. Stir in the chopped chives.
Transfer the cornmeal mixture to the prepared baking dish. Place in the oven for 30 to 40 minutes; or until puffed and lightly browned. Cool slightly before garnishing with chives and serving. This may also be baked in advance and reheated before serving.