Mash the avocado and the mayonnaise together in a bowl until thoroughly mixed.
Preheat a panini sandwich press.
To make the sandwiches, split the ciabatta rolls in half the flat way, and brush the bottom of each roll with olive oil. Place the bottoms of the rolls onto the panini press, olive oil side down. Place a provolone cheese slice, half the spinach leaves, half the sliced turkey breast, and a sliced roasted red pepper on each sandwich. Spread half of the avocado mixture on the cut surface of each top, and place the top of the roll on the sandwich. Brush the top of the roll with olive oil.
Close the panini press and cook until the bun is toasted and crisp, with golden brown grill marks, and the cheese has melted, about 5 to 8 minutes.
Instead of using a panini press, you can use a heavy cast iron skillet or griddle with a heavy weight (such as another heavy pan or even a brick wrapped in aluminum foil). Place the sandwich into the heated skillet or griddle, top with the heavy weight, and push down on the weight to press and crisp the sandwich. When the bread is toasted, flip the sandwich and press the other side.
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