For the chicken, remove the wings, legs and thighs from the chicken. Place all of the chicken pieces, including the crown, into a large pan with the honey mead and thyme sprig. Cover with a lid and bring to the boil, then reduce the heat and simmer for 25-30 minutes, or until the chicken is cooked through. Remove the chicken from the pan and pat dry, then carve the chicken breasts away from the crown. (You can reserve the chicken carcass for making stock in another recipe.)
Return the pan with the honey mead to the heat and bring to the boil, then simmer for 6-8 minutes, or until reduced slightly. Add half of the butter and season, to taste, with salt and freshly ground black pepper. Set aside.
Heat the remaining butter and olive oil in a frying pan until foaming, then fry the chicken pieces, skin-side down, for 2-3 minutes, or until golden-brown and crisp. Set aside and keep warm.
For the glazed carrots, cook the carrots in a pan of boiling salted water for 2-3 minutes, or until just tender. Drain and place into a clean pan along with the water, butter, honey and cumin, stirring well.
Bring to the boil, then cook for 4-5 minutes, or until the liquid in the pan thickens to a syrupy consistency and the carrots are glazed and tender. Season, to taste, with salt and freshly ground black pepper.
To serve, divide the chicken among four serving plates and spoon some glazed carrots alongside. Drizzle over the reduced honey mead and serve.