Step 1Place the diced carrot, shallots, celery, leek, lemon, bouquet garni and one-fourth cup olive oil in a heavy-bottom stock pot. Saut? over medium heat for 2 minutes until fragrant. Deglaze the pot with the white wine, chicken stock and 1 1/2 cups water. Add the artichoke hearts and bring to a boil. Reduce the heat and simmer, covered, for 20 minutes. Remove from the heat and strain the mixture, reserving the liquid. Set aside the vegetables (discard the lemon and bouquet garni) and liquid separately.
Step 2To make the barigoule, place the liquid in a large saucepan over medium heat and reduce by half. Dice two of the artichokes into one-fourth-inch pieces. Add artichokes and other vegetables to the reduced liquid. Remove from heat and let cool.
Step 3To make the artichoke foam, place the garlic with the remaining olive oil in a small saut? pan over low heat. Slowly cook until tender, careful not to let it burn, 5 to 7 minutes.
Step 4Place the reserved artichoke, the cream, garlic and one-fourth teaspoon salt and pepper in a medium saucepan. Simmer over medium heat for 4 minutes. Meanwhile, soften the gelatin sheet in a small bowl of cool water.
Step 5Remove the cream-artichoke mixture from the heat and pur?e in a blender or food processor until smooth. Pass the pur?e through a fine mesh strainer into a large bowl. Whisk the softened gelatin into the mixture until it is dissolved, and then gently fold in the whipped egg white. Place in the refrigerator, covered, for 2 to 3 hours until fairly stiff.
Step 1Sprinkle the abalone meat with the salt and let stand for 5 minutes. Using a large spoon and starting at the shallow end of the abalone, gently pry the meat from the shell. Clean the abalone: Using a kitchen towel, rub the salt into the abalone and scrape off the black sediment, then rinse with cold water. Using a sharp fillet knife, trim the hard "lip" at the edge of the abalone and carefully remove the "foot" (the tough portion that connects the abalone to its shell). Using the same knife, with a gentle sawing motion, carefully slice each abalone in half horizontally. Using the studded side of a tenderizing mallet, pound each slice on both sides until the flesh relaxes and is softened -- 1 to 2 good blows per side. Be careful not to over-pound and destroy the flesh.
Step 2Place the flour, eggs and bread crumbs into separate medium bowls. Bread each pounded abalone by first dredging it in flour, then egg, then the bread crumbs. Place the abalone on a parchment-lined baking sheet.
Step 3Place the saucepan with the artichoke barigoule over medium-low heat to warm. Stir in the butter and season to taste with salt and pepper. Remove from the heat but keep in a warm place.
Step 4Heat a cast iron or heavy-bottomed medium saut? pan over high heat. Add the clarified butter, swirling it to evenly coat the bottom of the pan, and heat until it is shimmering. Reduce the heat to medium and quickly add the breaded abalone, four fillets at a time. Fry the pieces until golden-brown and crisp on both sides, barely 20 seconds per side. Remove to a paper towel and season each piece with a light sprinkling of sea salt and pepper. Repeat until all of the abalone is fried.
Step 5To plate, place a spoonful of artichoke foam in the center of each of four plates. Place four pieces of abalone around the foam and spoon the barigoule around the abalone. Serve immediately.