In a medium sized pot over medium heat, melt the 6 tablespoons of butter and add the flour, stirring and cooking for about a minute, until it smells nutty.
Add about 2 tablespoons of milk at a time, whisking to combine. You should end up with a thick, velvety sauce.
Take the sauce off the heat and season with salt and pepper. Taste and adjust if needed.
Add the eggs, the cheese and the chives to the white sauce, and mix well to combine.
Pour into the buttered dish and bake for 30 to 40 minutes, or until the soufflé is puffy and well browned on top.