Brush polenta rounds with olive oil on both sides and put on a hot grill pan for about 3 minutes per side. Don’t rush this step; you want to get good grill marks or browning if you are using a sauté pan. Pat with paper towels to remove excess oil.
Put the parsley leaves and garlic into a mini food processor and process until finely chopped (or chop by hand).
Heat a large sauté pan over high heat. Add the 2 tablespoons of olive oil and heat until shimmering. Add the mushrooms and sauté until they start to brown and the liquid they put out is evaporated, about 2 or 3 minutes. Add the chopped parsley and garlic mixture and sauté for about 30 seconds. Add salt, pepper, and paprika. Add sherry and cook until all liquid is evaporated.