Preheat the oven to 400°F (200°C).
In a bowl, toss the halved baby potatoes with 1 tablespoon olive oil, 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper. Spread them on a baking sheet in a single layer.
Roast the potatoes in the preheated oven for 25-30 minutes, flipping 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 2 tablespoons 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 (internal temperature 165°F/74°C). Remove the 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 fresh peach puree, honey, apple cider vinegar, smoked paprika, and unsalted butter to the skillet. Stir and cook for 4-5 minutes until the sauce thickens slightly.
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 with the glaze.
While glazing the chicken, heat 1 tablespoon olive oil in another skillet over medium heat. Add the trimmed green beans, season with 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper, and sauté for 5-7 minutes until tender-crisp.
Remove the green beans from heat and toss with 1 teaspoon lemon juice.
To serve, plate the glazed chicken breasts alongside the roasted baby potatoes and sautéed green beans.