How to Make It
Preheat the oven to 350° and invert 4 small bowls on a work surface. Line a large rimmed baking sheet with a silicone baking mat or parchment paper. Sprinkle four 1/4-cup mounds of the aged Montasio cheese about 5 inches apart on the mat and spread into 4-inch rounds. Bake for 12 to 14 minutes, until golden and lacy. Let the frico cool for 1 minute and then, using a spatula, lay the warm frico over the bowls to cool and crisp. Transfer the cooled frico cups to a rack. Repeat with the remaining aged cheese to make a total of 8 frico cups.
In a large saucepan, bring the water and milk to a boil and add a generous pinch of salt. Whisk in the polenta in a slow, steady stream. Cook over moderately low heat, whisking frequently, until the polenta is thick and tender, about 30 minutes. Whisk in the young Montasio cheese and 2 tablespoons of the butter and season with salt and pepper; keep warm.
Meanwhile, in a medium skillet, cook the remaining 4 tablespoons of butter over moderate heat, swirling the pan, until the milk solids are golden and the butter smells nutty, about 4 minutes.
Transfer the frico cups to plates and fill with the polenta. Drizzle the brown butter on top and garnish with minced chives and shredded ricotta salata. Serve right away.