Heat olive oil in a large Dutch oven over medium-high heat. Add sliced shallots and 4 thinly sliced garlic cloves, season with salt and pepper. Cook, stirring occasionally, until totally softened and caramelized with golden-brown edges, 15 to 20 minutes.

Add red pepper flakes and anchovies. Stir until the anchovies dissolve into the shallots, about 2 minutes.

Add tomato paste, season with salt and pepper, and cook, stirring constantly, until the paste caramelizes at the edges and turns from bright red to a deeper brick red, about 2 minutes.

Cook pasta in salted boiling water until very al dente. Transfer pasta to the Dutch oven with 1 cup pasta water. Cook over medium-high heat, tossing constantly, until the sauce thickens and becomes sticky, 3 to 5 minutes.

Combine the finely chopped garlic clove and parsley in a small bowl, season with flaky salt and pepper. Divide pasta among bowls and top with the parsley gremolata and more red pepper flakes.