Preheat the oven to 375°F (190°C).
Place the turkey breast on a baking sheet. Drizzle with 1 tablespoon olive oil and sprinkle with 1/2 teaspoon salt and 1/4 teaspoon black pepper. Rub the seasoning evenly over the turkey.
Roast the turkey breast in the preheated oven for 50 to 60 minutes, or until the internal temperature reaches 165°F (74°C). Remove from oven and let rest for 10 minutes.
While the turkey roasts, place the diced potatoes in a medium pot and cover with water. Bring to a boil over high heat, then reduce to medium and simmer until potatoes are tender, about 15 minutes.
Drain the potatoes and return them to the pot. Add 1/4 cup whole milk, 2 tablespoons unsalted butter, and 1/2 teaspoon salt. Mash until smooth and creamy. Cover and keep warm.
Place the sliced carrots and 1/4 cup water in a small saucepan. Cover and steam over medium heat until tender, about 8 to 10 minutes. Drain any remaining water.
To make the gravy, melt 1 tablespoon unsalted butter in a small saucepan over medium heat. Whisk in 1 tablespoon all-purpose flour and cook for 1 minute, stirring constantly.
Gradually whisk in 1 cup low-sodium chicken broth. Continue whisking until the gravy thickens, about 3 to 5 minutes. Remove from heat.
Slice the rested turkey breast into kid-friendly pieces. Serve the turkey slices with mashed potatoes, steamed carrots, and drizzle with the homemade gravy.