Preheat the oven to 400°F (200°C).
In a medium bowl, toss the halved baby potatoes with 1 tablespoon olive oil, 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper. Spread the potatoes on a baking sheet in a single layer.
Roast the potatoes in the preheated oven for 25-30 minutes, turning once halfway through, until golden and tender.
While the potatoes roast, season both sides of the chicken breasts with 1 teaspoon kosher salt and 1/2 teaspoon black pepper.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the chicken breasts and sear for 4-5 minutes on each side until golden brown but not fully cooked through. Remove chicken from the skillet and set aside.
In the same skillet, reduce heat to medium and add the minced garlic. Sauté for 30 seconds until fragrant.
Add the peach preserves, Dijon mustard, apple cider vinegar, and smoked paprika to the skillet. Stir well and cook for 2-3 minutes until the glaze slightly thickens.
Return the chicken breasts to the skillet, spoon the peach glaze over them, cover, and cook for an additional 6-8 minutes until the chicken is cooked through and the glaze is sticky.
While the chicken finishes cooking, bring a medium pot of salted water to a boil. Add the green beans and cook for 3-4 minutes until tender-crisp. Drain and immediately transfer to a bowl of ice water to stop cooking.
Drain the green beans from the ice water and pat dry. In a small skillet, melt the butter over medium heat. Add the green beans and lemon juice, tossing to coat and warm through for 2 minutes. Season with a pinch of salt if desired.
To serve, plate the roasted baby potatoes alongside the peach glazed chicken breasts and sautéed green beans.