Put 3 tablespoons extra-virgin olive oil, 2 medium thinly sliced shallots and a pinch of salt in a large skillet and place over medium heat. Cook, stirring frequently, until the shallots are softened, but not browned.
Add 10-12 ounces portobello mushroom caps (sliced into chunks) to the pan. Cook the mushrooms for a few minutes until they take on some color, then stir and add ½ teaspoon salt. Continue cooking until the mushrooms become tender and their liquid evaporates.
Meanwhile, bring a large pot of water to a boil and add 2 tablespoons kosher salt. Cook the pasta until al dente, according to package directions — if using dried coiled pappardelle, add an extra minute or two and stir occasionally to help the strands separate, since they don't always unfurl as fast as the box says. Scoop out and reserve ¾ cup of the pasta water, then drain the pasta.
Add 2 tablespoons tomato paste, 2 finely sliced garlic cloves, 2 teaspoons finely chopped fresh rosemary leaves, 2 teaspoons balsamic vinegar, ½ teaspoon crushed red pepper, and 3 tablespoons unsalted butter to the mushrooms. Add ½ cup of the pasta water and stir over medium heat until the mixture becomes saucy. Add the pasta to the pan and toss gently with tongs to coat with the sauce, adding more water if it seems too dry. Taste for seasoning.
Shave a generous amount of Parmesan over the pasta with a vegetable peeler and serve.
Notes
Any leftover pasta and sauce will keep up to 3 days in the refrigerator.
The sauce coats up to 8 ounces of pasta. If you'd enjoy larger servings or are cooking one pound of pasta, add one more shallot and 1-2 additional portobellos. Increase the sauce by doubling the tomato paste, balsamic vinegar and butter. Add up to 1 cup of the pasta water.