In a large pot of boiling water, add pasta and cook until al dente.
While the pasta is cooking, start the mushrooms by heating a large frying pan over medium high heat with a tablespoon of olive oil. When the oil is hot, add the mushrooms.
Cook mushrooms until golden brown — 5-8 minutes. Then add diced shallots and garlic. Cook until fragrant, about one more minute.
Add the miso paste, evaporated milk, and parmesan, and stir until sauce is smooth.
By now the pasta should almost be done. Reserve 1 cup of the pasta cooking water, and then add the broccoli florets to the pot with the pasta. After a minute, the broccoli will be bright green and the pasta will be cooked. Drain and add pasta and broccoli to the frying pan with the mushroom mixture.
Toss to combine and emulsify the sauce. Add reserved pasta cooking water, a bit at a time, until sauce becomes smooth and clings to the pasta. Last, add lemon juice, lemon zest, chopped thyme and parsley, and the butter. Stir until smooth.
Divide pasta among 4 bowls. Serve with more parmesan cheese if desired.