Prepare the Filling:
Heat the olive oil in a cast iron skillet, oven proof pan or large saucepan. Add the chicken and fry on a medium heat for about 5 minutes or until the chicken is browned on all sides. Do not overcrowd the chicken - fry in batches if necessary. Do not stir too often, this may make the chicken release water.
When the chicken is browned, remove from the pan.
Reduce the heat to low, add the onions and fry for 3-5 minutes.
Add the carrots and celery and fry for a minute or so. Then add the garlic and continue frying for 1 minute.
Return the chicken to the pan and coat the meat and veg with the arrowroot. Cook gently for 1 minute.
Add salt, thyme, pepper, bay leaf and stock/broth. Bring to the boil, reduce the heat and simmer for 30 minutes.
While the stew is simmering, fry the mushrooms on medium high for 5 minutes with 1/2 tsp salt until they start to release their juices.
Add the fried mushrooms to the stew and continue cooking for 15 minutes.
If you didn't use a cast iron skillet or an oven proof pan, pour the chicken filling into a baking dish.
Prepare the topping:
While the filling is simmering, prepare the topping.
Preheat the oven to 200°C fan.
Heat the milk until lukewarm and add the vinegar to the milk so that the milk curdles. Leave to cool.
Mix together the flour, baking powder, salt, butter and herbs in your stand mixer with the paddle attachment, or by rubbing the mixture with your finger tips.
When the flour and butter mixture resembles bread crumbs add the milk mixture, sourdough discard. Mix until the dough only just comes together. Be careful not to over mix as this will make the topping tough. The dough will be quite dry, this is normal.
Turn out the dough onto a lightly floured surface and roll it out until it is about 2.5 cm (1 inch) thick. Using a round 6.5 cm (2.5 inch) cookie cutter cut the dough into rounds.
Assemble the cobbler:
Remove the bay leaf from the filling. Arrange the dough rounds on top of the hot chicken filling, brush with some extra milk or an egg wash (1 beaten egg diluted with a splash of water) and bake at 200°C fan for 20-25 mins or until the topping is golden brown.