Whisk together all of the ingredients for the vinaigrette in a small bowl and set aside.
PREPARE CHICKEN:
Sandwich the chicken breasts between 2 layers of plastic wrap and pound them very thin with the side of a meat cleaver or a rolling pin. Remove the chicken from the plastic wrap, cut into 4 chicken cutlets and season well on both sides with salt and pepper.
Heat a large sauté pan over medium heat. Unroll the pancetta so it looks like big strips of bacon. Add it to the pan and fry it like a tangle of bacon until the fat is rendered, 3 to 4 minutes. Drain on paper towels.
Drizzle 2 tablespoons of olive oil into the pan with the pancetta drippings. Add the chicken and pan-fry for 3 to 4 minutes on each side to brown the breasts and cook them through. Use a spatula to remove the chicken to the paper towels with the pancetta. Add the vinaigrette to the sauté pan and heat for a few minutes, stirring, to deglaze the bits of pancetta and chicken from the bottom of the pan; take the pan off the heat.
To serve, arrange the chicken on a platter. Scatter the blue cheese, arugula, figs, and pancetta over. Drizzle the dressing over everything and garnish with the tarragon leaves.