The large squares of pasta are meant to represent the swaddling clothes of the baby Jesus, but the dish is tasty even without any special meaning attached to it. What is key is the homemade pasta. And since the sauce takes no time at all, please take the time to make the pasta. (You will need a pasta maker for this recipe.)
In the bowl of a food processor, combine 2 cups flour, eggs, oil, and a large pinch of salt. Pulse and process until dough forms a ball. (If dough resists forming a ball, add a tablespoon or two of water.) Turn dough out onto a floured surface and knead a few times until smooth. Wrap dough in plastic wrap and let rest at least 30 minutes (refrigerate if letting rest longer). Line a sheet pan with a kitchen towel. Cut off one-third of dough and flatten it on a floured work surface. Place dough in pasta machine and roll at widest setting; fold in half and roll again. Continue folding and rolling at widest setting, rubbing dough with flour if it feels damp, until dough blisters. Roll (no folding anymore), lowering the setting with each roll and flouring as needed, to the second-to-last setting (number 6 on the Marcato Atlas pasta machine). Cut pasta into large squares using the width as your guide, and layer, flouring pasta squares well so they don’t stick together, on prepared pan. Repeat with remaining dough. Cover and refrigerate up to 3 days until ready to use. Bring a large pot of water to a boil. Salt water well and add pasta squares. Return water to a boil and cook pasta 1 minute, until just al dente. Drain and return pasta to pot. Add Anchovy Sauce, toss well, and serve immediately. Let guests garnish their pasta with cheese at the table.
In a small saucepan, combine all ingredients. Bring to a simmer over medium heat, swirling pan or stirring, until butter is bubbling and anchovies have dissolved into the sauce. Keep warm until serving.