Put the lemon slices on a tray and sprinkle both sides with salt and sugar. Transfer the slices to a rack and place in a dry area for 3 hours so the lemons can dehydrate. Rinse well in cold water and pat dry. Set aside.

Shuck the clams if you have purchased them in the shell. Slide an oyster knife into the unhinged side of each clam, carefully prying open the shells. Slice the knife around the margin of the clam meat, scraping the bottom of the shell to loosen the clam in 1 piece. Discard the shells. Slip off and discard the black sheath surrounding the siphon (the small "neck"). Rinse the shucked clams in running cold water.

Heat the peanut oil to 350 degrees F in a deep pot. Stir together the flour, salt, and peppers in a mixing bowl. Pour the buttermilk into a second, shallow bowl.

Fry the lemon slices first: dip each slice into the buttermilk and then into the seasoned flour. Fry the slices in small batches until they just turn golden brown, 1 to 2 minutes. Transfer to a tray lined with paper towels. The lemons can be fried 1 hour in advance and kept in a warm oven.

Remove any of the cooked lemon particles from the oil using a small metal strainer or a long-handled slotted spoon.

Dip each clam into the buttermilk and then into the seasoned flour. Place them in the hot oil individually (they tend to clump together). Cook in batches until crispy and brown, 3 to 4 minutes. Remove from the oil and transfer to a tray lined with paper towels. Sprinkle with salt and pepper, to taste.

Arrange the clams on a warm tray with a bowl of Tartar Sauce and the fried lemons.