How to Make It
Squeeze the lemons into a large bowl of water. Working with 1 artichoke at a time, snap off the outer leaves and trim off all but 1/2 inch of the stem. Using a sharp knife, cut off the leaves at the top of the heart. Peel the base and stem. Using a small spoon or a melon baller, scoop out the furry choke. Add the artichoke to the water and repeat with the remaining artichokes. Thinly slice the artichoke hearts and return them to the water. Drain and pat dry.
In a large, deep skillet, heat 1/4 cup plus 2 tablespoons of the oil until shimmering. Drain the artichokes and pat dry. Add the artichokes to the skillet and cook over high heat, stirring occasionally, until lightly browned in spots and barely tender, about 7 minutes. Add the garlic and cook, stirring, until lightly browned, about 2 minutes. Add the tomatoes and a generous pinch of salt and pepper and cook over low heat, smashing the tomatoes, until the artichokes are tender, about 5 minutes.
Meanwhile, in a large pot of boiling salted water, cook the pasta until al dente. Drain the pasta, reserving 3/4 cup of the cooking water.
Add the pasta and the cooking water to the skillet and simmer, tossing, until the liquid is nearly absorbed, about 2 minutes. Stir in the parsley.
In a large, nonstick skillet, heat the remaining 2 tablespoons of olive oil until shimmering. Season the fish with salt and pepper and add to the skillet, skin side down. Cook over high heat, turning once, until crisp and cooked through, about 5 minutes. Transfer the pasta to a large platter and arrange the fish on top, skin side up. Serve right away.