In a small saucepan over medium heat on the stove, combine the vinegar, dried figs, and shallot. Bring to a boil, reduce the heat to low, and simmer until almost all of the liquid has reduced to a thick, syrupy consistency, about 10 minutes. Remove from the heat and stir in the honey and lemon juice. Pour the glaze through a fine-mesh strainer into a glass or stainless steel bowl, and discard the pulp. Set aside to cool.
One at a time, place each breast, smooth side down, between two sheets of plastic wrap and pound to an even 12-millimeter thickness. Season both sides of the chicken with salt and pepper. Wrap one piece of prosciutto around the center of each breast, pressing the loose ends of the prosciutto down so the meat stays together. Lightly brush the chicken on both sides and the cut side of the figs with oil.
Prepare the grill for direct cooking over medium heat (180° to 230°C).
Prepare the grill for cooking over medium heat (200°C).
Brush the cooking grates clean. Grill the chicken, smooth (skin) side down first, over direct medium heat, with the lid closed, until the meat is firm to the touch and opaque all the way to the center, 6 to 8 minutes, turning once or twice. At the same time, grill the figs over direct medium heat until well marked and heated through, about 4 minutes, turning once after 3 minutes.
Brush the cooking grates clean. Grill the chicken, smooth (skin) side down first, over medium heat, with the lid closed, until the meat is firm to the touch and opaque all the way to the center, 6 to 8 minutes, turning once or twice. At the same time, grill the figs over medium heat until well marked and heated through, about 4 minutes, turning once after 3 minutes.
Divide the arugula evenly among four plates. Place one chicken breast on each serving of arugula along with four fig halves. Drizzle the chicken and arugula with the glaze and top with the cheese.
