Bring a large pot of salted water to a boil. Add the pasta and return to a boil, stirring occasionally. Cook until al dente according to the package instructions, 8 to 10 minutes. Reserve about 1 cup of the pasta water, then drain.
Meanwhile, add the olive oil to a large skillet and heat over medium heat. Add the garlic and shallot and saute until softened, about 2 minutes. Add the half-and-half and whisk in the goat cheese until smooth and warm. Season with pepper. Add the lemon zest and juice and half of the reserved pasta water and whisk until combined. Add half of the parsley, dill, mint and chives.
Transfer the drained pasta to the skillet and cook until the sauce starts to thicken, about 2 minutes. Turn off the heat, add 1/2 cup of the Pecorino Romano and stir to combine. Adjust the consistency of the sauce with the remaining pasta water if needed. Drizzle with extra-virgin olive oil.
Divide the pasta among bowls and garnish with the remaining Pecorino Romano, parsley, dill, mint and chives.