How to Make It
Put the chicken breast in a medium saucepan of boiling water, reduce the heat to low and simmer until cooked through, about 25 minutes. Transfer the chicken breast to a bowl and let rest for 15 minutes.
Meanwhile, in a large saucepan, cover the potatoes with water and bring to a boil. Salt the water and simmer over moderately high heat until tender, about 25 minutes.
In a small saucepan, cover the yellow pepper and jalapeño with water and bring to a boil. Simmer over moderately high heat until tender, about 8 minutes. Drain and transfer to a blender. Add the lime juice and puree.
When the potatoes are done, drain in a colander, then return to the saucepan. Shake the potatoes over high heat for about 15 seconds to dry them out. Using a potato masher, mash the potatoes well, then mash in the olive oil. Stir in the pepper puree and season with salt and pepper.
Discard the skin and bones from the chicken. Cut the chicken into 1/4-inch pieces and transfer to a bowl. Stir in the mayonnaise, celery and red onion and season with salt and pepper.
Line an 8-inch springform pan with plastic wrap. Spread half of the mashed potatoes in the bottom and partway up the side of the pan. Spread the chicken salad over the potatoes. Spread the remaining potatoes over the chicken and smooth the top. Cover with plastic wrap and refrigerate until lightly chilled, about 1 hour.
Carefully remove the ring from the pan and peel back the plastic wrap. Using 2 wide spatulas, transfer the cake to a serving plate. Smooth the side if necessary. Garnish the cake with the sliced avocado in a fan shape. Sprinkle the olives, cheese and sprouts over the top. Using a sharp knife, cut the cake into wedges and serve.