Want to snazz up your fried eggs? Cook them on top of melted parm, and you've got a frico egg!
Arrange a rack in the center of the oven; preheat to 450°. Line a large baking sheet with parchment paper.
In a large bowl, toss the mushrooms, oil, garlic, rosemary, crushed red pepper, cumin, dried thyme, allspice, and a generous amount of salt and coarsely ground pepper. Arrange the mushrooms on the baking sheet in an even layer. Roast for 10 minutes. Turn the mushrooms over and roast until browned, 10 to 12 minutes more.
If serving as an entrée, for the frico, heat a small (6-inch) nonstick skillet over medium. (If you have 2 skillets, you can make 2 frico at a time). Evenly scatter 1/4 to 1/3 cup cheese in the skillet. When it begins to bubble (1 to 2 minutes), crack an egg on top. Cover with a lid or foil. Cook until the white is set, about 2 minutes. Transfer to a plate. Repeat with the remaining cheese and eggs.
Meanwhile, in a medium saucepan, heat the stock over medium. When the mushrooms are about ready, add the polenta to the stock and whisk until thick, 2 to 3 minutes. Remove from heat. Add the butter and whisk until melted. Add any remaining cheese and the honey (if using). Season with salt and pepper. (If the polenta gets too thick, whisk in enough stock to make it spoonable.)
For a starter, swirl the polenta on small plates to create a well; fill with the mushrooms. Top with chives. For an entrée, serve the polenta and mushrooms topped with frico eggs and chives.