Pat the tomato slices dry with paper towels—this prevents excess moisture from making the bread soggy during grilling. Slice the avocado into thin, even pieces and set aside. Measure out the mayonnaise, oregano, and balsamic glaze into small bowls for easy access. Having everything prepped and ready ensures the panini comes together quickly once you start cooking.
Lay both bread slices on a clean work surface. Spread the mayonnaise evenly on the inside of one slice, then layer the fresh baby spinach over it. Arrange the patted-dry tomato slices on top of the spinach, overlapping them slightly. Top with the avocado slices and sprinkle the dried oregano over the vegetables. Place the cheese slices on top of the avocado. I like to use provolone for a more robust flavor, but mozzarella works beautifully too if you prefer something milder.
Close the panini by placing the second bread slice on top of the filling. Spread the softened butter evenly on the outside of both bread slices—this creates a golden, crispy exterior. Heat a skillet or panini press to medium heat (about 375°F if using a panini press, or medium on a stovetop). Once hot, carefully place the assembled panini on the grill surface. Cook for 3-4 minutes without moving it, allowing the bread to toast and the cheese to begin melting. Press gently with a spatula if using a skillet, or use the upper plate if using a panini press.
Carefully flip the panini (if using a skillet) or open the panini press and check that the first side is golden brown and crispy. Cook the other side for another 3-4 minutes until it's equally golden and the cheese has fully melted. I always let the panini rest for just a minute or two before cutting—this helps the cheese stay inside rather than oozing out immediately. Transfer to a cutting board, drizzle lightly with the balsamic glaze, then cut diagonally and serve immediately while warm.