In a large skillet, heat the olive oil over medium heat.
Add the diced onion and sauté for about 3 to 4 minutes, until the onion becomes translucent and soft.
Reduce the heat to low, then add the butter and garlic to the skillet.
Sauté for an additional 1 to 2 minutes, stirring intermittently until fragrant.
Pour in the cream, add ⅔ of the grated truffle, grated Parmesan, ½ teaspoon of salt, and pepper, and stir well to combine with the onions and garlic.
Let the sauce simmer gently over low heat for 5 to 7 minutes while the flavors meld together. Then, turn off the heat.
In the meantime, fill a large pot with water, bring it to a boil, and add 1 tablespoon of salt.
Add the tagliatelle to the boiling water and cook according to the package instructions until al dente, stirring occasionally.
Drain the pasta and reserve 1 ½ cups of pasta water.
Add the cooked tagliatelle to the truffle cream sauce along with 1 cup of pasta water.
Gently toss the tagliatelle in the sauce until it's well-coated. (If the sauce is too thick, you can add more pasta water to achieve the perfect consistency.)
Finish the truffle cream pasta with chopped parsley.
Divide the truffle pasta among serving plates or bowls and garnish with the remaining grated black truffle.