To cook the salmon, place the olive oil over a low heat and hold at a temperature of 40C.
Place the salmon on the oil for about 30 minutes, turning frequently until it is 37C in the middle (check this by inserting a skewer in the fish and touching it to your lip).
Meanwhile, heat a large sauté pan over a medium high heat and add 1 tablespoon of vegetable oil. Add the girolles along with a generous pinch of salt.
Sauté the girolles for 30 seconds then add the chicken stock and half of the butter. Cook for about 2 minutes until the girolles are tender.
Strain, keeping the cooking liquid as this will become the sauce.
Cook the carrot dice in a pan of salted boiling water until tender, drain and set aside.
To make the sauce, reduce the cooking liquid by three-quarters then whisk in the remaining butter. Continue to reduce until you have a nice sauce consistency.
Add the cooked girolles, carrots cut into 5mm dice, peas and lettuce. Cook until the lettuce has wilted. Season with a little lemon juice and pepper.
Portion the ragout into 4 bowls then carefully carve the salmon fillets in half and arrange on the top of each dish. Garnish with pea shoots.