Preheat the oven to 190°C/375°F. Have a shelf ready in the middle of the oven without any shelves above it. Arrange the celery, fennel and carrots into a trivet shape that the chicken will just cover nicely. Season the chicken cavity with a pinch of salt and plenty of pepper, then stuff with the lemon halves and thyme.
Sit the chicken on the vegetables and smother the breast and legs all over with the butter, then season the outside with a pinch of salt and plenty of pepper. Place in the oven and cook for 1 hour and 20 minutes until the chicken is cooked through and golden brown.
After 45 minutes, place the potatoes in a pan of cold water with a pinch of salt, then cover and bring to the boil. Reduce the heat and simmer for 7-8 minutes. Drain and leave in a colander to cool for 5 minutes, then give them a good shake to fluff up the edges.
Remove the chicken from the oven, quickly tip the potatoes into the tin and shake until evenly coated in the fat, using a spoon to help you if necessary. Then return the chicken to the oven to finish cooking. To check that it is fully cooked, pierce the thigh with a skewer and the juices should run clear. Transfer the chicken and roasted vegetables to a warm platter and cover loosely with tin foil. Leave to rest for 20 minutes. Give the potatoes another good shake and then return to the oven until crisp and golden brown.
Steam the green beans for 3-4 minutes and then season with a pinch of salt. Carve the chicken into slices and arrange on warmed plates with the roasted vegetables, steamed green beans and crispy roast potatoes to serve.