Bring a large pot of generously salted water to a boil.
While the water heats, place a small pan over medium heat. Add the olive oil and garlic cloves. Lower the heat and gently cook until the garlic softens and turns lightly golden, about 5 minutes. Remove from heat and set aside. Do not brown the garlic.
Once the water is boiling, add the kale leaves. Boil for about 5 minutes, until tender but still vibrant green.
Using tongs or a slotted spoon, transfer the kale directly to the blender. Do not discard the water. This will be used for the pasta.
Add the garlic and olive oil to the blender, along with a small ladle of the hot cooking water. Blend for about 1 minute until you have a smooth, bright green sauce. Taste and adjust with salt and freshly ground black pepper if needed.
Cook the pasta in the same water used for the kale, according to package instructions, until al dente. Reserve 1 cup of the pasta cooking water, then drain.
Return the pasta to the pot over low heat. Add the kale sauce, ¾ of the grated cheese and a small amount of pasta water.
Gently toss until all the pasta is evenly coated. Add more pasta water as needed until the sauce becomes loose, creamy and glossy.
Serve immediately, finishing each plate with the remaining cheese and a drizzle of olive oil.