Heat ¼ cup of the olive oil in a 2-quart saucepan over medium heat; add the garlic and cook until the garlic is fragrant, 1 to 2 minutes. Add the cauliflower florets, salt, pepper, chile flakes and ½ cup of the white wine. Top with an additional ¼ cup of olive oil; cover with a tight-fitting lid and simmer for 15 to 20 minutes, lowering the heat as needed to maintain a steady but gentle simmer. You do not want to brown the garlic. The cauliflower will steam in the wine and in its own water, which will slowly release.
Add additional wine in ¼-cup increments to keep the pot moist and cook until the cauliflower has become very tender. Use a potato masher to break up the cauliflower. Correct seasoning to taste with additional salt and pepper.
Meanwhile, cook the pasta according to package directions; drain, reserving 1 cup of the pasta cooking liquid.
Add the pasta back to the pot and mix with the cauliflower sauce and the cheese. Cook pasta and sauce over medium-low heat for several minutes, adding pasta cooking water as needed, and stirring often, until the pasta absorbs the sauce.