Position a rack in the lower third of the oven and preheat to 450 degrees F.
In a small bowl, combine the curry paste, vegetable oil, brown sugar and 1 teaspoon of kosher salt.
Put the green beans and carrots on an oiled sheet pan and pour slightly less than half the curry mixture over them. Use a pair of tongs to toss and coat them. Season with 3/4 teaspoon of salt.
Nestle the chicken drumsticks on the sheet pan. Using a spatula or spoon, spread the remaining curry mixture on the drumsticks. Do your best to coat them evenly. Season the chicken with the remaining 3/4 teaspoon salt.
Roast the chicken and vegetables (turning them halfway through to prevent them from getting too dark on the bottom) until they are tender and a thermometer inserted into the thickest part of the chicken (without touching the bone) reads 170 degrees F, about 30 minutes.
Scrape the chicken, vegetables, and any browned bits from the pan onto a large serving dish and garnish with the cilantro.