Cook the pasta. Bring a large pot of water to a boil, then add about 1 Tbsp kosher salt. Add tagliatelle, then cook according to package directions, or until al dente. Reserve about ½ cup of pasta water, then drain pasta and set aside.
Begin making the sauce. Heat 2 Tbsp olive oil and 2 Tbsp butter in a large skillet over medium-high heat. When hot, add 3 cloves chopped garlic, 2 anchovies, and 1 tsp red pepper flakes, then use a wooden spoon to break down the anchovies (they should melt into the sauce), cooking for about 2-3 minutes.
Finish the pasta. Add crab and cook until just heated through, about 1 minute, then deglaze with ⅓ cup white wine. Season with 1 tsp Kosher salt, then cook 1 minute more. Add drained pasta, lemon juice, lemon zest, ¼ cup chopped parsley, ¼ cup chopped dill, 1 Tbsp drained capers, and and ¼ cup parmesan. Stir to combine.
Add cream, then season. Pour in about ¼ cup of reserved pasta water and ¼ cup of heavy cream, then toss gently. If you want more of a sauce, add more reserved water and/or cream, otherwise taste and adjust seasoning as needed. Garnish with additional black pepper, Parmesan, and chopped parsley, then serve immediately.