Heat oil in a large, heavy-bottomed saucepan over medium heat. Saute onion, garlic, carrots, mushrooms, summer savoury, thyme, parsley, salt, and pepper, stirring often, until onions are softened.
Add flour and cook for 1 minute. Slowly whisk in the milk and chicken stock, then bring to a boil.
Add green beans. Reduce heat and simmer for 5-8 minutes until mixture is thick enough to coat the back of a spoon. Stir in chicken.
Transfer chicken mixture to a 2-litre baking dish; set aside.
To make the biscuit topping, add flour, baking powder, and salt to a food processor. Pulse in butter until crumbly. Slowly add in milk, pulsing until a sticky dough forms. You may need just ? of a cup, you may need more.
Transfer dough to a floured surface and knead a few times until smooth. Shape into a disk wide enough to cover your baking dish. Cut into 4 wedges, then place wedges on top of the chicken mixture.
Bake for 35 minutes until biscuits are browned and no longer doughy on the bottom.