Bring a large pot of salted water to a boil. Meanwhile, cook the garlic in the olive oil in a large skillet over medium-high heat, stirring occasionally, until golden brown and crisp, about 3 minutes. (Be careful not to over-brown the garlic or it will taste bitter.) Using a slotted spoon, transfer the garlic chips to a paper-towel-lined plate. Pour off all but 2 tablespoons oil into a small bowl to use as a dip for crusty bread, if desired. Add the onions and red pepper flakes to the oil in the pan; cook, stirring, until the onions are light brown, about 10 minutes. Season with 1 1/2 teaspoons Kosher salt.
When the onions are almost done, add the greens to the boiling water and cook, uncovered, until just tender, about 2 minutes. Using tongs, remove the greens, shaking off the excess water; add them to the skillet with the onions (set the pot of water aside). Cook, stirring occasionally, until tender, about 5 minutes.
Return the cooking water to a boil. Add the spaghetti and cook until al dente, 8 to 10 minutes. Remove and reserve about 1 cup cooking water; drain the pasta and transfer to a serving bowl. Add the cheese and toss. Add the greens and some of the reserved pasta water and toss, adding more water as necessary to keep the pasta from clumping. Top with the garlic chips.