Peel and finely chop the onion. Trim the carrot and finely chop it to match the onion (no need to peel). Peel and finely chop the potato.
Warm a large pan over a medium heat for 1 min, then add 1 tbsp olive oil and the chopped veg. Season with a pinch of salt and pepper. Gently cook for 5 mins, stirring occasionally, till the veg have softened but not picked up too much colour. If they start to brown, turn the heat down.
While the veg cook, thickly slice the mushrooms. Quarter or halve any small mushrooms. Stir them into the pan with the veg and cook for another 5 mins till the mushrooms look juicy.
Crumble in the stock cube and add 1 tsp caraway seeds. Add the pearl barley and bay leaf. Pour in 800ml boiling water and give everything a stir to mix it well. Pop on a lid and simmer the soup for 20 mins, till the barley is nice and tender.
While the soup simmers, roughly chop the soft dill fronds. Squeeze the juice from half the lemon into the soup. Taste it and add a pinch more salt or pepper, or a squeeze more lemon juice, if you think it need sit. Ladle the soup into warm bowls and garnish with chopped dill to serve.