For the Chicken and Vegetables: Adjust oven rack to upper-middle position and preheat oven to 450°F (230°C). In a large bowl, stir to combine curry paste, oil, salt, fish sauce, and black pepper. Add chicken, carrots, and broccoli, and use hands to toss to coat evenly. On a rimmed sheet pan, place chicken thighs, skin side up, 1 thigh in each corner. Reserve carrots and broccoli in the bowl; set aside.

Bake chicken for 10 minutes. Carefully remove pan and add carrots in space around chicken, in a single layer. Bake until carrots begin to soften, 10 to 15 minutes. Add broccoli to sheet pan in a single layer around carrots and chicken, arranging florets cut-side down, if applicable. Bake until vegetables are tender-crisp and a thermometer inserted into thickest portion of chicken registers at least 175°F (80°C), 5 to 10 minutes. (If needed for browner skin, increase oven temperature to broil without removing baking sheet and broil chicken until skin is crisped and tips of broccoli florets begin to char, 2 to 3 minutes. ) Let rest for 10 minutes before serving.

For the Cilantro-Peanut Relish: While chicken and vegetables rest, in a small bowl, stir together cilantro, peanuts, chile, lime juice, and fish sauce. Set aside until ready to use.

For the Peanut-Lime Dressing: In a small saucepan, whisk together coconut milk, peanut butter, lime juice, curry paste, fish sauce, and sugar. Bring to a simmer over medium heat, whisking often, until smooth and thickened, about 1 minute.

To serve, divide chicken and vegetables evenly among 4 plates. Drizzle with warm peanut-lime dressing, and top with cilantro-peanut relish. Serve immediately.