Preheat a panini press. (You may use a skillet or grill pan if you don’t have a panini press.)
In a small bowl, mash the avocado, mayonnaise, salt and pepper together until thoroughly mixed.
Brush one side of each piece of bread with olive oil, and place with the olive oil side down.
Top two of the bread slices with three pieces of the thin sliced cheese or two pieces of the regular, breaking as needed to evenly cover the bread.
Place half of the sliced turkey, half of the spinach leaves, and half of the sliced pepper on top of each piece of cheese covered bread.
Spread half of the avocado mixture on the two remaining slices of bread (on the side without the olive oil), and place on top of the turkey covered pieces.
Carefully, place the sandwiches on the panini press, and cook until the bread is golden brown and crisp and the cheese has melted, about 3 to 5 minutes.
Cut in half and enjoy warm.