Roast the shallots. Preheat oven to 425 F. Pour 2 Tbsp olive oil on a rimmed baking sheet, a large cast iron skillet, or roasting pan. Place shallots on top, cut side down. Drizzle with another 1 Tbsp olive oil, then season with ½ tsp Kosher salt. Roast for 25 minutes.
Bring water to a boil. Bring a large pot of water to a rapid boil. Season generously with Kosher salt (a good 2 Tbsp).
Add pasta. Place bucatini or other pasta in the water, then stir gently to separate. Cook for 7 minutes, or until al dente according to package directions.
Reserve cooking liquid. Reserve 1 cup of pasta cooking water in a measuring cup, then strain pasta.
Make the crunchy breadcrumbs. Meanwhile, melt 2 Tbsp butter and 2 Tbsp extra virgin olive oil in a large skillet over medium-high heat. When hot, add 4 cloves finely chopped garlic, 4 whole anchovies, and 1 tsp lemon zest. Stir with a spatula, breaking up the anchovies so they melt into the sauce. Once dissolved, add ½ cup breadcrumbs, then stir until toasted and all the liquid has been absorbed, about 1-2 minutes. Breadcrumbs should be light golden brown and evenly toasted.
Add pasta to the pan. Pour the cooked pasta into the skillet with the toasted breadcrumbs. Add the caramelized shallots, ¼ cup reserved pasta cooking water, ¼ cup Parmigiano Reggiano, remaining 2 Tbsp butter, 2 Tbsp fresh lemon juice, and 2 Tbsp chopped parsley. Use tongs to mix everything together. Taste, then adjust seasoning as needed. If you want a looser, thinner sauce, add more reserved pasta water a little at a time.
Finish and serve. Transfer the shallot pasta to a large serving bowl or individual plates. Top each with additional chopped parsley or parmesan, as desired. Serve with a drizzle of high quality extra virgin olive oil and a generous pinch of flaky sea salt. Enjoy immediately!