Chicken Fricassee (Fricassee De Poulet a L'Ancienne)

INGREDIENTS

• 1 whole chicken (3 1/2 to 4 pounds), cut into 11 pieces (reserve back and wing tips for another use, such as stock)
• Coarse salt and freshly ground pepper
• 3 tablespoons unsalted butter, softened, divided
• 1 tablespoon extra-virgin olive oil
• 1 small yellow onion, cut into 1/4-inch dice (1 cup)
• 1 carrot, cut into 1/4-inch dice (1/2 cup)
• 1 celery stalk, cut into 1/4-inch dice (1/3 cup)
• 8 ounces cremini mushrooms, trimmed and quartered
• 2 tablespoons all-purpose flour
• 2/3 cup dry white wine
• 4 cups chicken broth
• 2 sprigs fresh flat-leaf parsley
• 2 sprigs fresh thyme
• 1 bay leaf
• 2 large egg yolks, room temperature
• 1/4 cup heavy cream
• 2 to 3 tablespoons roughly chopped fresh tarragon leaves
• 2 tablespoons fresh lemon juice