Place chicken in a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until no longer pink in the center, about 20 minutes. Drain and cool until easily handled.

Bring another pot of water to a boil while chicken is cooking; boil potatoes and carrot in their skins until easily pierced with a fork, about 20 minutes. Drain and cool until easily handled.

Cut cooled chicken into thin strips. Peel and grate potatoes and carrot.

Line a 9-inch pie plate with plastic wrap. Arrange mushrooms stem-side up over the plastic wrap. Sprinkle spring onions on top. Dot a little mayonnaise all over. Layer carrot over spring onions. Layer 1/2 the potatoes over the carrot. Dress with some mayonnaise.

Arrange eggs, Cheddar cheese, and cornichons in layers over the mayonnaise. Dress with more mayonnaise. Cover with chicken strips, remaining potatoes, and remaining mayonnaise. Cover with plastic wrap and refrigerate for 2 hours.

Uncover and invert pie plate onto a serving platter so mushrooms are cap-side up. Peel off the plastic wrap before serving.