Bring a large pot of water to a boil over high heat.

Meanwhile, in a medium saucepan over medium heat, warm the oil until shimmering. Add the garlic and tomato paste and cook, stirring, until incredibly fragrant and the tomato paste has darkened slightly in color, about 2 minutes. Add the vodka and bring the mixture to a boil. Let it boil vigorously to cook off the harshness of the vodka, less than 1 minute. Add the crushed tomatoes and heavy cream, reduce the heat to low and season to taste with salt. Cover the saucepan to prevent splattering, and let the sauce simmer while you cook the pasta.

Generously salt the boiling water, add the pasta and cook it for 2 minutes less than the package instructions for al dente. (It finishes cooking in the sauce.) Transfer 1 cup of the pasta cooking water to the sauce mixture, then drain the pasta and return it to the now-empty pot. Pour the vodka sauce over the pasta and add the spinach. Turn the heat to low and cook, stirring to combine the pasta with the sauce and spinach, until the pasta is al dente and has absorbed some of the sauce, and the spinach is warmed through, about 2 minutes.

Divide among bowls, top with cheese and serve hot.