Position a rack in the center of the oven and heat the oven to 450°F.
Whisk the olive oil, vinegar, lemon juice, and 2 tsp. of the garlic in a large shallow bowl. Add the chicken, turn to coat, and let sit at room temperature, turning occasionally, for 30 minutes.
Put the chicken skin side up on a rack set over a large rimmed baking sheet (reserve the marinade). Season well with salt and pepper. Roast for 10 minutes, lower the heat to 375°F, and roast until the skin is deep golden and the meat is cooked through (165°F), about 30 minutes more.
Meanwhile, melt the butter in a 12-inch heavy-duty skillet over medium heat. Add the reserved marinade and the shallots. Cook, stirring, until tender and lightly browned, 3 to 4 minutes.
Add the chanterelles and cook, stirring frequently, until tender and golden-brown on some sides, about 3 minutes. Add the remaining garlic and cook, stirring, until fragrant, about 1 minute. Add the wine and simmer for 1 minute. Remove from the heat. Set aside 1/2 cup of the mushrooms and cover to keep warm.
Transfer the chicken to a plate, reserving the juice in the baking sheet and scraping up any browned bits. Put the skillet with the mushroom mixture over medium-high heat and add the chicken drippings, cream, peas, 3/4 tsp. salt, and 1/2 tsp. pepper. Cook until thickened to a sauce consistency, about 2 minutes. Remove from the heat, add 2 tsp. of the tarragon, and season to taste with salt and pepper. Divide the sauce among four shallow bowls and top with the chicken. Sprinkle with the reserved mushrooms and the remaining tarragon, and serve.