Step 1Combine the Dijon and whole-grain mustards, the vermouth, oil, tarragon and egg. Marinate the chicken by slathering it with the mustard mixture in a shallow glass dish. Make sure to get the marinade under the skin. Cover and refrigerate overnight.
Step 1Heat the oven to 375 degrees.
Step 2Toss the bread with the olive oil on a baking sheet and toast until golden brown, about 5 minutes. The croutons should be crisp but still tender inside.
Step 1Heat the butter in a skillet over medium-low heat until it foams. Add the shallot wedges, thyme and salt and pepper and cook until the shallots are softened and a deep caramel brown, about 15 minutes. Set aside.
Step 2Heat the olive oil in a large skillet over medium heat and add the fennel wedges. Season with salt and pepper and sear the fennel gently. As the fennel begins to brown, turn the wedges over and scatter in the sliced shallots. Cook until the shallots are tender, 2 to 3 minutes. Add the vermouth and cook until reduced by half, about 5 minutes. Add the chicken stock and simmer until the fennel is tender, 5 minutes. Add the reserved shallots and the mustard greens and stir until the greens have wilted, about 5 minutes. Add the croutons and toss to coat them with fennel and shallot juices.
Step 3Meanwhile, heat the grill to medium heat. Grill the chickens, skin side down, until the skin is brown and bubbly, about 10 minutes. Turn the chickens over and cook the other side until the chicken is cooked through and the juices run clear when the chicken is pierced with a fork, 15 to 20 minutes.
Step 4Divide the warm bread salad mixture among large dinner plates. Place the chicken on top. Serve immediately.