For the mushy peas, drain the peas and rinse well under running water. Place the peas into a large, lidded saucepan, cover with 600ml water and bring to the boil. Reduce the heat until the mixture is simmering, cover the pan with a lid, then simmer for 25-30 minutes, or until tender. Remove the lid and continue to cook until most of the water has evaporated and the peas are very soft.
Drain off any excess water from the cooked peas, then add the butter and lemon juice and mix well. Season, to taste, with salt and freshly ground black pepper.
For the beer and dill-battered cod, sprinkle the yeast and sugar into a mixing bowl. Pour over the beer and vinegar, mix well, then whisk in the flour and salt until the mixture is well combined.
Set the mixture aside to ferment. (It is ready to use when the mixture starts to bubble.)
When the mixture has fermented, add the dill and stir until well combined.
Heat the oil in a deep heavy-based frying pan until a breadcrumb sizzles and turns brown when dropped into it. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Dip the cod into the batter, then fry in the oil for 3-5 minutes, or until golden-brown and crisp and cooked through. Remove the fish from the oil using a slotted spoon and set aside to drain on kitchen paper. Season, to taste, with salt and some of the lemon juice.
Mix the olive oil and a teaspoon of the remaining lemon juice in a bowl until well combined. Add the pea shoots and mix until well coated in the dressing
To serve, spoon some of the mushy peas into the centre of four serving plates, then top with a piece of battered cod and a handful of pea shoots.