Bring a large pot of salted water to a boil. Cook tagliatelle until al dente, approximately 3 minutes less than package instructions.
While the pasta is cooking, heat olive oil in a large saute pan over medium-high heat. Add the garlic and red pepper flakes and cook until fragrant. Add the lemon juice, zest and approximately 1/4 cup of the pasta water.
Transfer the pasta to the saute pan. Add the butter then toss to coat the pasta. Continue cooking until the sauce is thickened, approximately 3 minutes more. Season with salt and freshly ground black pepper.
Remove pan from heat and stir in Parmigiano-Reggiano while tossing constantly.
Divide among 4 bowls or plates.