Preheat the oven to 350°. Lightly brush the baguette slices with the 2 tablespoons of olive oil and arrange on a baking sheet. Toast the slices for 15 minutes, or until golden and crisp.
Meanwhile, in a small saucepan, cover the beet with cold water and bring to a boil. Cook over moderate heat until tender when pierced with a knife, about 20 minutes. Drain and let cool. Peel the beet and cut it into 1/4-inch dice.
In a medium bowl, combine the onion with 2 teaspoons of the vinegar and a generous pinch each of salt and pepper and let stand for 20 minutes. Stir in the beet, tomato, radicchio and 1 tablespoon of the oil. Let the beet salsa stand at room temperature for 1 to 3 hours.
In a large, heavy skillet, heat 2 tablespoons of the olive oil until shimmering. Add the chicken livers, season with salt and pepper and cook over high heat until browned, about 1 minute; transfer to a plate. Add the remaining 1 tablespoon of olive oil to the skillet along with the ginger and cloves and cook until fragrant. Add the apple juice and the remaining 1 teaspoon of red wine vinegar and cook, scraping up the browned bits from the bottom of the pan, until thickened, about 3 minutes. Return the chicken livers to the skillet, season with salt and pepper and toss gently until the livers are just rosy within, about 1 minute longer. Cut the chicken livers into 1 1/2-inch pieces.
Arrange the crostini on a large platter and top with the chicken livers. Spoon the pan drippings over the livers, then carefully spoon the beet salsa on top and serve immediately.
A fruity Pinot Gris makes a fine contrast to the rich chicken liver.