Using high-quality extra virgin olive oil and cheese is mandatory in such a simple dish. Toast walnuts in a dry skillet over medium-low heat, stirring often to prevent burning. Use more garlic, if you want.
- Warm olive oil and butter in a medium-sized skillet over medium-low heat. Add garlic, and cook slowly until garlic is fragrant and just starting to turn golden, about 10 minutes. Stir in parsley and toasted walnuts, and keep warm over very low heat.
- Bring large pot of lightly salted water to a boil. Place tagliatelle into water, and cook according to package directions. Drain, and toss pasta gently with warmed olive oil mixture.
- Add Parmigiano-Reggiano, salt and generous grindings of black pepper, and toss until strands are well coated. Serve immediately in warmed pasta bowls.