In a medium bowl, whisk the flour and lemon juice together to make a paste, then whisk in 6 cups cold water. One at a time, cut away the stems of the artichokes and rub each bottom with a lemon half Remove and discard the tough outer leaves until those exposed are folded inward and are creamy green in color. Cut off the cone of leaves halfway down from the top and discard. Rub the cut surface with the used lemon half and drop into the water-flour mixture.

Fill a large nonreactive pot with lightly salted water and bring to a boil. Squeeze the remaining lemon half into the boiling water. Add the drained artichokes, reduce the heat, and simmer about 30 minutes until the bottoms are tender and can be easily pierced with a sharp paring knife. Remove the artichoke bottoms with a slotted spoon to a colander, cool under cold running water, and drain. (The hearts can be prepared one day ahead: prepare up to this point, cover, and refrigerate.)

Carefully remove the remaining leaves, disturbing the bottom as little as possible. Scrape out the choke with the tip of a spoon. Trim the sides to make a neat shape. Shave the bottoms flat so they will sit level. Cover and keep warm, or reheat in a steamer over boiling water.

In a medium saucepan of lightly salted boiling water, cook the carrots until tender, about 6 to 8 minutes. Drain, and purée in a food processor. Season with salt and pepper and keep warm in a double boiler over simmering water.

Prepare the beurre blanc: In a nonreactive medium saucepan, boil the orange juice and white wine over medium heat until the liquid is syrupy and reduced by two-thirds, about 12 minutes.

Remove the pan from the heat and whisk in 2 tablespoons of the butter, one piece at a time. Return to low heat and, whisking constantly, add the remaining butter, one piece at a time. When all of the butter is added, the sauce will have the creamy smooth consistency of a hollandaise. Remove the pan to the warm side of the stove to keep warm.

Bring a large saucepan of water to a simmer. Add the vinegar. One at a time, carefully break each egg into the water. Using a large spoon, quickly life the setting white around each egg and spoon it towards the egg yolk to help the eggs poach into oval shapes. Simmer gently until the whites are firm, but the yolks are still soft, about 4 minutes. Use a slotted spoon to transfer the eggs to paper toweling to drain briefly. Trim away some of the whites so the eggs will sit neatly on top of the artichoke hearts. Set the cooked poached eggs aside in a bowl of ice water until ready to serve, up to 2 hours. (You should have extra eggs on hand just in case any should break.)

Bring a skillet of plain water to a simmer. With a slotted spoon, transfer the poached eggs to the water and heat through, about 2 minutes. Place the artichoke hearts on individual warm plates and fill the wells with a spoonful of the carrot purée. Use the slotted spoon to place the eggs on top of the purée. Spoon the beurre blanc over the eggs and sprinkle with chives.