Combine the dried porcini mushrooms and water in a small microwave safe bowl. Cover with plastic wrap, cut several steam vents, and microwave on high for 30 seconds. Let stand until the mushrooms soften, about 5 minutes. Lift the mushrooms from the liquid with a fork and mince. Pour the liquid through a small strainer lined with a single sheet of paper towel into a measuring cup.
Meanwhile, combine the white mushrooms, shallot, 1 tbls. of the oil, and 1/2 teaspoons salt together in a 12" nonstick skillet. Cover and cook over medium low heat until the mushrooms release their juice, about 7 minutes.
Uncover, increase the heat to medium high, and stir in the garlic and thyme. Continue to cook, uncovered, until the mushrooms are golden brown, about 5 minutes. Off the heat, stir in the strained porcini water, scraping up any browned bits.
Process the sauteed mushroom mixture, minced porcini, Parmesan, ricotta, parsley, and remaining 1 tbls. oil in a food processor until smooth. Season with salt and pepper to taste.
When tossing the pesto with pasta, add some of the pasta cooking water as needed to loosen the consistency of the pesto.
The pesto can be covered with a sheet of plastic wrap pressed flush against its surface and refrigerated for up to 3 days. Do not freeze this pesto.