Bring a medium pot of salted water to a boil. Place walnuts in a sealable bag; use a meat mallet, heavy skillet, or rolling pin to pound until very finely crushed. Finely grate Parmesan. Halve and thinly slice 2 tablespoons shallot; finely chop the remaining. Stack pasta sheets, then cut lengthwise into 1-inch wide strips. Cover with a damp paper towel to keep from drying out.
In a medium bowl, whisk to combine 2 tablespoons oil, 1 tablespoon vinegar, and a pinch each of salt and pepper. Stir in cherries and sliced shallots, then set aside to marinate, stirring occasionally.
Heat 1 tablespoon butter in a medium skillet over medium-high. Add chopped shallots and cook, stirring, until softened, about 2 minutes. Add crushed walnuts and cook, stirring constantly, until golden brown and fragrant, about 2 minutes (watch closely.) Transfer mixture to a small heatproof bowl, then season with salt and pepper. Wipe out skillet; reserve for step 5.
Add pasta to boiling water and cook, stirring to prevent clumping, until al dente, 2–3 minutes. Reserve 1 cup cooking water, then drain pasta.
Add pasta, mascarpone, walnut mixture, half of the Parmesan, ¾ cup reserved cooking water, and ⅛ teaspoon pumpkin pie spice to reserved skillet. Simmer over medium-high until sauce is slightly thickened and pasta is coated, about 2 minutes (sauce should be slightly loose, as pasta will continue to soak up liquid). Season with salt and pepper. Stir in more cooking water, if necessary.
Add arugula to bowl with dressing and toss to coat. Season to taste with salt and pepper. Serve pappardelle with walnut cream sauce topped with remaining Parmesan, and with arugula salad alongside. Enjoy!