Heat your oven to 190°C/Fan 170°C/Gas 5. Put a large pan of salted water on to boil.
Finely grate the Parmesan onto a plate. Pick the leaves off the thyme sprigs, finely chop them and add them to the Parmesan with the breadcrumbs. Stir together to mix. Sprinkle 2 tbsp of the flour onto a separate plate and season with salt and pepper. Crack the egg onto a third plate and beat with a fork.
Line a baking tray with baking paper. Pull the skin off the chicken thighs, slicing it off if necessary. Dip the chicken thighs into the flour, then the egg and then the Parmesan breadcrumbs, turning to make sure they’re well coated. Place them on the baking tray and bake for 15 mins.
While the chicken thighs bake, pull the skins off the drumsticks and dip in the flour, egg and breadcrumbs. Leave them on the breadcrumb plate till the thighs have baked for 15 mins, then add them to the baking tray and bake the chicken for 30 mins. The thighs will cook for 45 mins, the drumsticks for 30 mins. They should be golden brown when done.
While the chicken cooks, scrub the potatoes and chop them into equal bite-sized pieces. When the water in the pan is boiling, add the potatoes. Simmer for 15 mins till the potatoes are tender when pierced with a fork.
While the potatoes simmer, trim the roots and top 1cm off the spring onions and finely slice them. Pop them in a bowl. Trim the dry ends off the celery and finely slice it. Add it to the bowl.
In a small bowl, combine the crème fraîche and 1 tbsp Dijon mustard. Add a pinch of salt and pepper and stir together.
Drain the potatoes. Let them steam dry in the colander for a few mins, then add to the spring onions and celery. Stir in the mustardy crème fraîche dressing.
When the chicken is ready, lift it off the tray and divide between 2 warm plates. Serve with the warm potato salad and handfuls of the lamb’s lettuce on the side.