How to Make It
In a large skillet, cook the carrots with the sugar and 2 tablespoons of the vinegar over moderate heat, tossing, until barely softened, 2 minutes. Season with salt and pepper. Spread on a plate and refrigerate until chilled.
Wipe out the skillet and heat 1 tablespoon of the oil in it. Add the beets and onion and cook over moderate heat for 1 minute. Add the remaining vinegar and 1/4 cup of water, cover and cook until the beets are crisp-tender and dry, 5 minutes. Spread the beets and onion on a plate, season with salt and pepper and refrigerate until chilled.
Preheat the oven to 350°. Light a grill. In a bowl, whisk the mayonnaise and horseradish. Brush the chicken with the remaining oil and season with salt and pepper. Grill over high heat, turning once, until lightly charred, 5 minutes. Transfer to a cutting board and let cool slightly. Cut crosswise into strips.
Toast the baguettes in the oven for 5 minutes, until crusty. Split and cut each baguette into 4 lengths. Spread the mayonnaise on the bread and top with the beets and onion. Top with the chicken, carrots and lettuce. Close the sandwiches and serve.