Pat the outer surface of the chicken dry with paper towels. In a small bowl combine 1 teaspoon of salt, ½ teaspoon of pepper and ½ teaspoon of the thyme. Use that mixture to season the chicken outside and inside the cavity. Let the chicken stand at room temperature for 30 minutes before grilling.
Prepare the grill for indirect cooking over medium–high heat (about 200ºC).
Pour enough water into the centre cup of the poultry roaster to fill it about one-quarter of the way (but no more). Place the chicken over the centre cup so it sits firmly. Tuck the tips of the wings behind the chicken’s back.
Place the bell peppers and zucchini in a large bowl. Drizzle with 2 tablespoons of oil and season with ½ teaspoon of salt, the remaining ½ teaspoon of thyme and ¼ teaspoon of pepper. Toss to coat.
Arrange the vegetables around the perimeter of the poultry roaster. Cook the vegetables with the chicken over indirect medium–high heat, with the lid closed, until an instant-read thermometer inserted into the thickest part of a thigh (not touching the bone) reaches 70ºC to 74ºC, and the vegetables are tender, 1 to 1¼ hours.
Wearing insulated barbecue mitts, remove the poultry roaster from the grill and transfer the chicken to a cutting board. Let rest for about 15 minutes (the internal temperature will rise a few degrees during this time).
Carve the chicken into serving pieces and arrange on a serving platter. Spoon the vegetables around the chicken. Garnish with tarragon. Serve warm.