Preheat oven to 350°F. Generously grease a 2-quart casserole dish with butter. Heat milk and buttermilk in a medium-size heavy saucepan over medium until steaming but not boiling. (Look for small bubbles forming around edges of saucepan.) Gradually pour cornmeal into milk mixture in a slow, steady stream, whisking or stirring with a large spoon. (Scrape sides of saucepan as you whisk to prevent lumps from forming.) Reduce heat to medium-low. Cook, stirring often, until cornmeal dissolves completely and mixture thickens, 10 to 12 minutes.

Remove cornmeal mixture from heat. Add butter, chopped thyme, salt, and pepper, stirring until butter melts and mixture is combined. Let stand 10 minutes.

Place egg yolks in a medium bowl. Stir in 2 cups of the cornmeal mixture until combined. Transfer cornmeal-egg yolk mixture to remaining cornmeal mixture in saucepan, stirring quickly until combined. (Mixture may still look a little grainy or lumpy.)

Beat egg whites in a medium bowl with an electric mixer fitted with a whisk attachment on medium-high speed until stiff peaks form, about 1 1/2 minutes. Gently fold egg whites into cornmeal mixture using a spatula or large spoon until mixture becomes a smooth batter with no streaks remaining.

Spoon batter into prepared casserole dish. Bake in preheated oven until spoon bread is puffed up, firm, and lightly browned, 35 to 40 minutes. (Note that spoon bread will initially puff up to great heights but then deflate within minutes of leaving the oven.) Serve hot or warm.