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1
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.
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2
Season with salt and pepper.
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3
Spread on a plate and refrigerate until chilled.
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4
Wipe out skillet and heat 1 tablespoon of the oil in it.
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5
Add the beets and onion and cook over moderate heat for 1 minute.
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6
Add the remaining vinegar and 1/4 cup of water, cover and cook until the beets are crisp-tender and dry, 5 minutes.
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7
Spread the beets and onion on a plate, season with salt and pepper and refrigerate until chilled.
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8
Preheat the oven to 350.
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9
Light a grill.
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10
In a bowl, whisk the mayonnaise and horseradish.
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11
Brush the chicken with the remaining oil and season with salt and pepper.
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12
Grill over high heat, turning once, until lightly charred, 5 minutes.
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13
Transfer to a cutting board and let cool slightly.
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14
Cut crosswise into strips.
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15
Toast the baguettes in the oven for 5 minutes, until crusty.
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16
Split and cut each baguette into 4 lengths.
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17
Spread the mayonnaise on the bread and top with the beets and onion.
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18
Top with the chicken, carrots and lettuce.
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19
Close the sandwiches and serve.
