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Preheat oven to 425F.
Butter a nonstick, regular-size muffin tin.
Bring the milk and stock to a bubble in a medium-sized stockpot over medium heat. Slowly whisk in the polenta and continue whisking until it begins to thicken and resemble hot oatmeal, about 3-4 minutes. Turn off the heat, add in the cheese and season with salt and pepper.
While the milk and stock are coming up to a bubble, place the sausages and grapes on a baking sheet. Drizzle them with EVOO and season with pepper. Pour the balsamic over everything and pop the baking sheet into the oven for 20-25 minutes.
Pour the polenta into the muffin tin, halfway up each muffin cup. Once all the cups are filled, form a well with the back of a spoon by pushing the polenta up the sides until it reaches the top rim (dip the spoon in water so it won't stick to the polenta). The well will be relatively large but that's OK -- you want the egg to have a little nesting place so it won't overflow out of the tin.
Once you have made all the wells, crack an egg into each of the polenta-filled cups.
Bake in the oven for 20 minutes, until firm (the yolk will still be a little runny). Remove from the oven and let sit.
In a medium-sized serving bowl mix together the arugula, basil, and lemon juice. Drizzle with some EVOO and season with a little salt and pepper. To serve the polenta egg cups, run a small butter knife around the edges of the polenta to loosen it from the sides of the muffin tin. Gently pry them out with a fork or a knife and serve with the sausages and salad alongside.