Bring a large pot of well-salted water to a boil over high. Set a large nonreactive skillet over medium heat, and add 3 tablespoons oil. When oil shimmers, add the onion and a pinch of salt. Cook, stirring occasionally, until the onion is tender and just beginning to brown, about 7 minutes.

Move the onion to the edges of the skillet, and add 2 tablespoons oil and the garlic. Cook, stirring constantly, until fragrant, about 1 minute. Add the anchovies, chile paste to taste, and tomato paste; cook, gently breaking up the paste with a wooden spoon, until the oil turns uniformly orange and the anchovies dissolve, about 2 minutes.

Add the pasta to the boiling water, and cook until nearly al dente, 7 to 12 minutes, depending on the shape. Reserve 1 cup pasta water, and drain the pasta.

Increase the heat under the skillet to medium, and add the cherry tomatoes. Ladle a little reserved pasta water into the cherry tomato can, swirl it around, and add it to the pan. Bring the sauce to a simmer over medium, using the back of the spoon to break up the tomatoes. Cook, stirring occasionally, until the tomatoes begin to appear jammy, 4 to 6 minutes.

Stir in the Parmigiano-Reggiano, tuna, and tuna oil from can. Taste the sauce, and, taking into consideration that you’ll still be adding some salted pasta water, adjust the seasoning with salt and chile paste as desired.

Increase the heat under the skillet to medium-high, and add the drained pasta to the skillet, tossing with the sauce until well coated. As the sauce continues to cook and thicken onto the noodles, add pasta water, a few tablespoons at a time, to loosen it up as much as needed. The sauce may also need another tablespoon or so of olive oil. This last step is all about tasting, stirring, and tinkering until the sauce comes together to coat the pasta in just the right way, ideally at precisely the same moment the noodles cross over to al dente.

Grate the zest of the lemon over the pasta, and add the parsley. Toss again; taste and add lemon juice, if desired. Serve immediately, topped with more chile paste and Parmigiano-Reggiano, if desired.