MAKE THE GNOCCHI: In a medium saucepan, melt 1 tablespoon of the butter. Add the bacon, carrot, and onion and cook over moderate heat, stirring, until the onion is softened, about 4 minutes. Add the lentils and water, cover and simmer over low heat until the lentils are tender, about 30 minutes. Uncover and boil over high heat until most of the water has evaporated, about 5 minutes.

Set aside 1/4 cup of the lentils. Transfer the remaining lentils and any liquid to a food processor and puree. Pass the puree through a coarse strainer into a large bowl; you should have about 1 cup.

Bring a large saucepan of salted water to a boil, then reduce the heat so the water simmers. Set a large bowl of ice water nearby. Add the flour, cheese, egg, egg yolk, salt, and pepper to the lentil puree and stir until a dough forms. Transfer the dough to a resealable plastic bag. Using scissors, cut a 1/2-inch diagonal opening from a corner of the bag. Working over the simmering water, pipe out and cut 1/2-inch pieces of the dough, letting them drop into the water. Cook the gnocchi until tender, about 4 minutes. Using a slotted spoon, transfer the gnocchi to the ice water to cool. Drain, gently pat dry with paper towels and transfer to a bowl.

In a large nonstick skillet, melt the remaining 2 tablespoons of butter. Add the gnocchi and cook over moderate heat, stirring a few times, until lightly browned, about 3 minutes.

PREPARE THE BALSAMIC ONION WEDGES: Put the onion wedges on a microwave-safe plate and drizzle with the white balsamic vinegar. Cover with plastic wrap and microwave on high power for 1 minute.

MAKE THE FOIE GRAS BROTH: In a medium saucepan, boil the port over high heat until reduced by half, about 4 minutes. Add the chicken stock and bring to a boil. Pour the mixture into a blender and add the foie gras cubes. Puree until smooth, then pass the foie gras broth through a fine strainer into the saucepan. Warm the broth over moderately low heat, whisking constantly; do not boil. Season with salt and pepper.

Rewarm the gnocchi. Spoon the gnocchi and the reserved lentils into 6 small bowls. Ladle the foie gras broth over the gnocchi and garnish with the onion wedges. Drizzle each serving with aged balsamic vinegar and serve right away.

The boiled gnocchi can be lightly tossed with oil and refrigerated overnight.