How to Make It
Sprinkle 1 tablespoon salt all over chicken. Place chicken on a wire rack set inside a rimmed baking sheet. Chill, uncovered, 8 hours.
Melt butter in a large skillet over high until sizzling. Add half of chanterelles in a single layer, and cook until lightly browned on bottom, about 3 minutes. Stir and cook, stirring occasionally, until browned all over, about 8 minutes. Reduce heat to medium; add shallot and garlic, and cook, stirring constantly, until slightly softened, about 1 minute. Stir in white wine, sherry, tarragon, 4 thyme sprigs, and bay leaves. Bring to a strong simmer over medium-high; reduce heat to medium, and gently simmer, stirring occasionally, until liquid is slightly reduced, 5 to 6 minutes. Stir in cream, and return to a simmer over medium. Simmer, stirring occasionally, until liquid is slightly thickened (bubbles will get larger as sauce reduces), about 5 minutes. Remove from heat; remove and discard bay leaves, tarragon sprigs, and thyme sprigs. Place mushroom mixture in a blender, and process until smooth, about 1 minute, stopping to scrape down sides as needed. Spoon pureed chanterelle sauce into a small saucepan. Set aside.
Preheat a gas grill to medium-high (400°F to 450°F) on one side, or push hot coals to one side of a charcoal grill. Place chicken, breast side down, on a cutting board. Using poultry shears, cut along both sides of backbone, separating backbone from chicken; remove backbone. (Discard or reserve for stock.) Turn chicken breast side up. Press firmly against breastbone until it cracks. Tuck wing tips under wings. Rub chicken with 1 tablespoon garlic oil. Place chicken, breast side down, on oiled grates over lit side of grill. Grill, covered, rotating chicken occasionally and flipping once to color evenly, until chicken is lightly browned, about 15 minutes. Move chicken to unlit side of grill, breast side up. Grill, covered, until a thermometer inserted in thickest part of breast registers 155°F, 20 to 25 minutes. Transfer chicken to a work surface, and tent with aluminum foil.
Increase grill heat or add coals as necessary to bring heat to high (450°F to 500°F). Toss together 1 tablespoon garlic oil, 1 teaspoon salt, and remaining half of chanterelles in a large bowl. Toss together nectarines, remaining 1 tablespoon garlic oil, and remaining 1/2 teaspoon salt in a medium bowl. Place chanterelles on oiled grates (or place in a grill basket) over lit side of grill, and grill, turning often, until mushrooms are tender and browned in spots, 4 to 8 minutes, removing mushrooms from grill as they finish cooking. Transfer mushrooms to a medium bowl, and toss with lemon juice and remaining 4 thyme sprigs. Season with salt to taste. Place nectarines, cut sides down, on oiled grates over lit side of grill, and grill, uncovered, until slightly softened, about 2 minutes per side. Remove from grill.
Reheat chanterelle sauce over medium, stirring occasionally, and spread on a large serving platter. Cut chicken into 8 pieces, and place on platter. Add grilled nectarines and chanterelles, spooning mushroom liquid from bowl over platter. Garnish with chives, nasturtium leaves, and bronze fennel fronds.