Cook Torchiette according to package directions and drain, reserving 1/2 cup pasta cooking water.
In a large skillet, heat oil over medium heat. Add anchovies, garlic, capers, and red pepper flakes. Cook and stir until anchovies dissolve, 2 to 3 minutes.
Add pasta and lemon juice, stir to coat, adding pasta water as needed if mixture is too dry. Remove from heat and add eggs, Parmesan cheese, and parsley. Toss to combine. Transfer to serving bowl and serve.