Preheat the oven to 375°F (190°C). Pat the chicken thighs dry and season both sides with salt and pepper.
Heat 1 tablespoon olive oil in an oven-safe skillet over medium-high heat. Add the chicken and sear 5 minutes per side until browned. Remove the chicken to a plate.
Reduce heat to medium and add 2 tablespoons butter to the skillet. Add the sliced shallot and cook 2 minutes until softened, then add the minced garlic and cook 30 seconds.
Stir in 3/4 cup peach preserves, 1/4 cup chicken broth, 2 tablespoons honey, 1 teaspoon Dijon mustard, and 1 teaspoon dried thyme. Whisk and cook about 2 minutes until the sauce begins to thicken.
Return the browned chicken to the skillet, nestling it into the sauce.
Arrange the sliced peaches over the chicken, then transfer the skillet to the preheated oven and bake 15–20 minutes, until the chicken reaches 165°F (74°C) and the peaches are tender.
Remove from the oven, garnish with fresh basil leaves, and serve.