Step 1
Pulse 6 oz. shiitake mushrooms, stems removed, in a food processor until very coarsely chopped (about pea-size pieces). Transfer to a large bowl. Add 1 small beet, scrubbed, chopped, and its greens, 1 large shallot, and 3 garlic cloves to processor and pulse until coarsely chopped; add to bowl. Add one 15-oz. can chickpeas, rinsed, patted dry, to processor and pulse until the consistency of coarse breadcrumbs. Transfer to bowl and add 1 tsp. Aleppo-style pepper or other mild chile flakes and 2 Tbsp. white miso; mix until combined.
Step 2
Heat ⅓ cup vegetable oil in a large heavy skillet over medium-high. Scrape mushroom mixture into skillet; reserve bowl. Cook, stirring occasionally, until most of the moisture is evaporated and mixture is charred in spots and almost crumbly, 15–20 minutes. Taste and season with kosher salt if needed. Return mixture to reserved bowl; reserve skillet.
Step 3
Cook 12 oz. spaghetti or other long pasta in a large pot of boiling salted water until very al dente, about 3 minutes less than package directions.
Step 4
Meanwhile, thinly slice remaining 2 garlic cloves. Heat 2 Tbsp. extra-virgin olive oil and 2 Tbsp. unsalted butter in reserved skillet over medium until butter is melted. Cook garlic and remaining 2 Tbsp. miso, stirring often, until garlic is golden and miso is slightly darkened in color and beginning to stick to bottom of skillet, about 2 minutes.
Step 5
Just before pasta is al dente, scoop out 1½ cups pasta
cooking liquid and add to skillet. Return mushroom mixture to skillet and stir to
combine, scraping up any browned bits; reduce heat to low.
Step 6
Using tongs, transfer pasta to skillet. Add 1 oz. Parmesan, finely grated (about ½ cup), and season with freshly ground pepper. Toss to combine and coat pasta, adding up to ½ cup more pasta cooking liquid as needed to form a silky sauce. Taste and season with more salt if needed.
Step 7
Divide pasta among shallow bowls and season with more pepper. Top with more finely grated Parmesan.