Mash avocado and mayonnaise together in a bowl until thoroughly combined.

Preheat a panini sandwich press.

Brush bottom of each roll with olive oil; place onto the panini press, oiled sides down. Place 1 slice cheese, 1/2 spinach leaves, 1/2 turkey, and 1/2 sliced peppers on each roll on panini press. Spread 1/2 avocado mixture on cut surface of each top roll; place 1 top roll on each sandwich. Brush top of each roll with olive oil.

Close the panini press; cook until rolls toasted and crisp, with golden brown grill marks, and the cheese has melted, 5 to 8 minutes.