Pat the chicken completely dry with paper towels and remove any giblets from the cavity. Season the cavity and outside of the chicken with the salt and black pepper.
In a small bowl, stir together the softened butter, honey, French seasoning, garlic, Dijon mustard, and lemon zest until smooth. Carefully loosen the skin over the breasts and thighs using your fingers. Spread about two-thirds of the butter mixture underneath the skin, then rub the remaining butter over the outside of the chicken.
Pour the chicken broth into the Instant Pot and place the trivet inside. Arrange the lemon halves and rosemary sprigs on the trivet, then place the chicken breast-side up on top.
Secure the lid and set the valve to Sealing. Cook on High Pressure for 6 minutes per pound (see Notes for cooking chart).
Allow a 15-minute Natural Release, then carefully release any remaining pressure. Check that the breast reaches 165°F and the thighs reach 175 to 185°F.
Arrange an oven rack in the center position and preheat the broiler. Carefully transfer the chicken to a foil-lined baking sheet. Brush with an additional drizzle of honey if desired for extra color, then broil for 4 to 8 minutes, watching closely, until the skin is golden brown and lightly crisp.
Let the chicken rest for 10 minutes before carving and serving.
For optional gravy, strain the cooking liquid into a saucepan or use the Instant Pot on Sauté mode. Whisk together the cornstarch and cold water until smooth, then stir into the simmering drippings until thickened. Season with additional salt and pepper to taste.
