Fill a large lidded pot halfway with generously salted water and set on the stove to boil.
In a wide skillet, heat the olive oil over medium heat. Add the onion and garlic, sprinkle with ¼ teaspoon of the salt, and cook, stirring occasionally, until softened.
Meanwhile, cook the cauliflower in the pot of boiling water until tender, then remove with a slotted spoon.
Return water to a boil and cook the pasta until al dente.
Add the shiitakes and remaining ¼ teaspoon salt to the skillet with the onions and cook, stirring frequently, until lightly browned and cooked through. Use a metal spatula to lift them from the bottom of the pan if they start to stick.
Pour the capers and brine into the skillet and cook for a moment, using the brine to loosen any browned bits from the bottom of the pan.
Add the chickpeas and cook until warmed through.
Drain the pasta and add to the skillet along with the cauliflower, lemon juice, pepper, and grated cheese.
Serve with additional grated cheese.