Bring a large pot of water to a rolling boil. Salt it generously. See Note #4 below.
Cook the pasta according to package directions until 1 minute shy of al dente. Before draining, scoop out at least ¾ cup of pasta water and set it aside.
While the pasta cooks, heat the olive oil in a large skillet over medium-low heat. Add the shallots and cook, stirring occasionally, for 3 to 4 minutes until soft and translucent.
Add the garlic and red pepper flakes and cook 1 minute more until fragrant. (Keep the heat medium-low. You want everything to soften, not brown.)
Add the anchovy fillets to the pan. Use the back of a wooden spoon to press and smear them into the oil. They will completely dissolve in about 2 minutes.
Add the tomato paste and stir it into the oil. Let it cook and caramelize for 60 to 90 seconds, stirring frequently, until the color deepens and the aroma changes.
Pour in the white wine and let it bubble up. Let it reduce by about half, around 2 to 3 minutes.
Add the drained pasta directly to the skillet along with a generous splash of the reserved pasta water and most of the parsley. Toss well.
Add the cold butter one or two cubes at a time, tossing constantly until the sauce is silky and glossy
Add the lemon zest and a good squeeze of lemon juice. Taste and adjust salt and black pepper.
Serve and garnish with the additional parsley. Finish with freshly grated Parmigiano Reggiano or Pecorino Romano.