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

Add red-pepper flakes and anchovies. Stir to melt the anchovies into the shallots, about 2 minutes.

Add tomato paste and season with salt and pepper. Cook, stirring constantly to prevent any scorching, until the tomato paste has caramelized a bit, about 2 minutes. Remove from heat and transfer about half the mixture to a resealable container.

Cook pasta according to package instructions in a large pot of salted boiling water until very al dente. Transfer to Dutch oven with remaining shallot mixture and 1 cup pasta water. Cook over medium-high heat, swirling to coat each piece of pasta until sauce has reduced, about 3 to 5 minutes.

In a small bowl, combine parsley and finely chopped garlic clove, season with flaky salt and pepper. Divide pasta among bowls and top with parsley mixture and more red-pepper flakes, if desired.