Lower the heat to medium and cook, stirring constantly, until the polenta is very thick, about 15 minutes. Season with 1 teaspoon kosher salt and 1/2 teaspoon freshly ground black pepper.
Divide the polenta in half and place in two bowls. To the first bowl of polenta, add 1 tablespoon butter, cut into small pieces, and 1/4 cup (1 oz) grated Parmesan cheese. Stir well.
To the second bowl of polenta, add 1 tablespoon butter, cut into small pieces, and 1/4 cup roasted red pepper puree. Stir well. I roasted one red bell pepper, took the charred skin off, and pureed it in my blender. If you have not previously roasted a pepper, check out my how-to on the subject.
Line a baking sheet with plastic wrap. Pat the Parmesan polenta into one corner of the baking sheet. The square should be about 1/2-inch thick. Pat the red pepper polenta beside the Parmesan polenta. They should be equal in height. If the polenta sticks to your fingers as you’re patting it, dampen your fingers slightly.
The basil sauce:
Now, you can easily make this sauce in a food processor in about 30 seconds flat. However, I was feeling rather self-righteous and decided to make the sauce (really a pesto) with a mortar and pestle. I was channeling my inner Italian nonna. This method give the pesto a much different texture than when you puree it.
Chop 2 cloves garlic and place it in the mortar with 1 teaspoon kosher salt.
Place one eggplant round on each red pepper polenta round.