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 until evenly coated.
Spread the potatoes on a baking sheet in a single layer and roast in the preheated oven for 25-30 minutes, turning 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 until shimmering.
Add the chicken breasts to the skillet and cook for 5-6 minutes on one side without moving them, until golden brown.
Flip the chicken breasts and cook for another 5-6 minutes on the other side, or until the internal temperature reaches 165°F (74°C). Remove the chicken from the skillet and cover loosely with foil to rest.
In the same skillet, reduce heat to medium, add 3 tablespoons unsalted butter and the minced garlic. Cook for 1-2 minutes, stirring frequently, until the garlic is fragrant and the butter is melted.
Add the green beans, 1/4 teaspoon kosher salt, and 1/8 teaspoon black pepper to the skillet with the garlic butter. Toss to coat and cook for 5-7 minutes, stirring occasionally, until the green beans are tender but still crisp.
Add 1 tablespoon unsalted butter to the green beans and stir until melted and evenly distributed.
Serve the rested chicken breasts alongside the roasted baby potatoes and garlic butter green beans.