While chicken is roasting, heat remaining 1 tablespoon olive oil in a small saucepan over high heat until shimmering. Add chicken backbone and breastbone and cook, stirring frequently, until well browned, about 3 minutes. Add onion, carrot, and celery and cook, stirring frequently, until beginning to brown, about 3 minutes. Deglaze with vermouth and 1 cup water, using a wooden spoon to scrape up any browned bits from bottom of pan. Reduce heat to maintain simmer and cook for 30 minutes. Strain out solids and return liquid to pan. Boil over medium-high heat until approximately ⅓ cup remains, about 7 minutes. Whisk in soy sauce, butter, and lemon juice off heat. Season to taste with salt and pepper.