Mascarpone Polenta with Brown Butter Mushrooms

Preheat oven to 450°F with racks in middle and bottom third of oven.

Bring 5 cups water, 1/4 cup butter, garlic, 2 1/4 teaspoons salt, and 1/2 teaspoon pepper to a boil in a large saucepan over medium-high. Remove and discard garlic. Reduce heat to low to maintain a gentle simmer. Whisking constantly, gradually add polenta in a slow stream. Return to a gentle simmer over low, stirring often. Cook, stirring and scraping bottom of saucepan often using a wooden spoon, until polenta is tender and creamy, 25 to 30 minutes. Remove from heat, and stir in mascarpone cheese; season with salt and pepper to taste. Cover and keep warm.

Meanwhile, toss together mushrooms, 1/4 cup oil, 1/2 teaspoon salt, and remaining 1/2 teaspoon pepper in a large bowl. Divide mushrooms between 2 large rimmed baking sheets, and spread evenly. Roast in preheated oven until mushrooms are crisp in spots and tender, about 25 minutes, rotating baking sheets between top and bottom racks and stirring halfway through roasting time.

Melt remaining 3/4 cup butter in a large nonstick skillet over medium-high, about 2 minutes. Continue to cook, stirring often, until milk solids turn brown and smell nutty, about 2 minutes. Carefully scrape butter into a medium-size heatproof bowl. Set aside 1/3 cup brown butter. Whisk apple butter and remaining 1 1/4 teaspoons salt into remaining brown butter in medium bowl.

Wipe skillet clean. Heat remaining 2 tablespoons oil in skillet over medium-high until shimmering. Crack eggs into skillet; season with salt and pepper to taste. Cook until whites are set and yolks are still runny, spooning hot oil over any egg whites that remain translucent, about 2 minutes. Remove from heat.

Divide polenta evenly among 4 bowls. Add mushrooms and herbs to apple butter mixture in bowl; toss to combine. Top polenta evenly with mushroom mixture and fried eggs; drizzle evenly with reserved brown butter. Garnish with additional herbs. Serve immediately.