How to Make It
In a medium bowl, whisk the egg yolks, lemon zest, 1 tablespoon of the lemon juice and 1 1/2 teaspoons of the fine salt. Whisking constantly, slowly stream in the olive oil until the aioli is thick and glossy. Whisk in 1 more tablespoon of the lemon juice and the anchovies. Cover and refrigerate.
Add the remaining 3 tablespoons of lemon juice to a large bowl of cold water. Working with 1 artichoke at a time, trim the stem. Snap off the leaves until you reach the tender light green inner leaves. Cut off the top third of the artichoke and trim off any tough leaves near the base. Halve the artichoke lengthwise and scoop out the fuzzy choke if necessary. Drop the artichoke in the lemon water. Repeat with the remaining artichokes.
In a medium, straight-sided skillet, heat 2 inches of canola oil to 250°. Line a rimmed baking sheet with paper towels and top with a rack. Drain the artichokes well and pat dry. Fry in 3 batches over moderately high heat until tender and just beginning to brown, about 5 minutes. Using a slotted spoon, transfer the artichokes to the prepared wire rack to drain.
Heat the oil to 375°. Fry the artichokes again in 3 batches until crispy, about 1 minute per batch. This time, drain on the paper towels. Season generously with sea salt and serve hot with the aioli and lemon wedges.