Preheat the oven to 190°C.
In a large bowl, combine the biscuit ingredients: flour, baking powder, thyme, and salt. Add cold, diced butter and rub it into the flour until the mixture resembles coarse crumbs. Pour in the milk and mix quickly until just combined – visible bits of butter are fine. Shape into a ball, wrap in cling film, and refrigerate for 30 minutes.
Peel the potatoes, carrots, and celery root, cut into pieces, and cook in salted water or steam until just tender (al dente). Drain and set aside. Cut the courgette and broccoli into small pieces – no need to pre-cook them.
In a large pot, melt the butter and add the finely chopped onion. Cook over medium heat until softened and translucent. Sprinkle in the flour, stir, and cook for another minute. Gradually pour in the cream, milk, and broth, whisking constantly to avoid lumps. Simmer over low heat, stirring often, until the sauce thickens slightly.
Add all the vegetables and shredded chicken to the sauce. Season with thyme, salt, and black pepper to taste. Transfer the filling to an ovenproof dish.
Take the dough out of the fridge, roll it out on a floured surface to about 1 cm thickness, and cut out round biscuits. Arrange them on top of the casserole and brush with beaten egg. Bake for 30–35 minutes, until the biscuits are golden and the filling is bubbling hot.