The milk helps give this polenta a creamy texture and wonderfully rich taste. Use a nonstick pan when frying the polenta rounds.
Lightly butter a 9-by-13-inch baking dish. Set aside. In a large saucepan, combine milk, water, and salt. Bring to a boil and slowly sprinkle in cornmeal, whisking constantly to break up lumps. Cook over low heat, stirring constantly with a wooden spoon, for 15 minutes.
Stir in butter, cheese, and pepper to taste. Pour into prepared dish, cover with plastic wrap, and refrigerate until firm (polenta can be refrigerated for up to 2 days).
In a nonstick frying pan over medium heat, melt a small amount of butter. Using a 3-inch round cookie cutter or a large glass, cut out 10 circles of polenta. Cook until brown on both sides, about 3 minutes per side. Transfer to a baking sheet and keep warm in oven on lowest setting.
In a large saucepan, bring about 3 inches of lightly salted water to a boil, then turn down to a bare simmer. One at a time, carefully crack eggs into a small shallow bowl, then gently slide into water. Poach two or three at a time for 3 to 5 minutes. Remove with a slotted spoon and drain well. Continue until all eggs are poached.
On serving plates, top each polenta round with a poached egg; spoon a little Herb Beurre Blanc on top, sprinkle with chopped chives, and serve immediately.