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Step 1
Position a rack in the lower third of the oven and preheat to 450 degrees F.
Step 2
In a small bowl, combine the curry paste, vegetable oil, brown sugar and 1 teaspoon of kosher salt.
Step 3
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.
Step 4
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.
Step 5
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.
Step 6
Scrape the chicken, vegetables, and any browned bits from the pan onto a large serving dish and garnish with the cilantro.
Step 7
Serve with rice, yogurt and lime wedges.