Preheat the oven to 400°F. Butter a 10-inch cast iron skillet or similar oven-safe baking dish.
On the stovetop in a medium-sized saucepan, whisk together the milk, salt, and cornmeal. Cook over medium heat, stirring constantly for about 3-5 minutes, until the mixture thickens to a pudding-like consistency. Remove from heat and let cool for 5 minutes
Add 2-3 Tbsp warm cornmeal mixture to the eggs to temper. Stir well.
Stir the cream cheese into the remaining cornmeal mixture in the saucepan.
Whisk the tempered eggs into the cornmeal. Mix until fully combined.
To the mixture add the chives and 1 1/2 cups shredded cheese. Spoon into the baking dish. Top with the remaining 1/2 cup shredded cheddar cheese. Dot with butter.
Bake for 25 to 30 minutes. The top and sides will be browned and the center will be just set.