Heat ¼ cup oil in a large Dutch oven pot over medium-high heat. Wait until the oil is nice and hot. Throw in the mushrooms, but do not overcrowd to avoid steaming. Cook in batches if need be. Toss the mushrooms occasionally until browned and tender, 5–8 minutes.
Reduce heat to medium-low. Add shallots and thyme, season with salt and pepper, and cook stirring often until shallots are translucent and softened about 2 minutes. Add in the garlic and miso paste and marry them well into everything in the pot.
Add stock and reduce heat to low. Bring to a simmer and cook until only a thin layer of stock coats the bottom of the pot, about 7 minutes.
While the mushroom base is simmering, cook pasta in a large pot of boiling salted water, occasionally stirring to avoid lumping, until very al dente, about 3 minutes less than package directions.
Using tongs, transfer pasta to pot with mushrooms and add 1 cup pasta cooking liquid. Increase heat to medium and bring to a simmer. Cook, tossing constantly, until pasta is al dente and liquid is slightly thickened (about 2 minutes).
Add quark, return to a simmer, and cook, tossing until pasta is coated, about 1 minute (try to avoid going longer otherwise the quark will curdle too much).
Remove from heat, add butter, and toss to combine.Top with more thyme, fresh black pepper, and grated pecorino romano.