Preheat the oven to 400°F (200°C).
In a large bowl, toss the halved baby potatoes with 2 tablespoons olive oil, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Spread them evenly on a baking sheet.
Roast the potatoes in the preheated oven for 25-30 minutes, stirring halfway through, until golden and tender.
While the potatoes roast, season the chicken breasts with 1 teaspoon salt and 1/2 teaspoon black pepper on both sides.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat.
Add the chicken breasts and cook for 5-6 minutes on each side until golden brown and cooked through. Remove chicken from skillet and set aside.
In the same skillet, combine the peach preserves, Dijon mustard, apple cider vinegar, garlic powder, and crushed red pepper flakes. Stir and cook over medium heat for 2-3 minutes until the glaze is warm and slightly thickened.
Return the chicken breasts to the skillet and spoon the peach glaze over them. Cook for an additional 2 minutes, turning the chicken to coat evenly in the glaze.
Meanwhile, bring a pot of salted water to a boil. Add the green beans and cook for 3-4 minutes until crisp-tender. Drain and immediately transfer to a bowl of ice water to stop cooking. Drain again.
In a medium skillet, melt 1 tablespoon butter over medium heat. Add the minced garlic and sauté for 30 seconds until fragrant.
Add the drained green beans and 1/4 teaspoon salt to the skillet. Toss to coat and heat through for 2 minutes.
Serve each chicken breast topped with peach glaze alongside roasted baby potatoes and sautéed green beans.