Halve, pit, and scoop the avocado into a bowl. Mash with a fork until mostly smooth, leaving a few small chunks for texture. Season with salt and a few grinds of black pepper.

Spread the mashed avocado evenly on two slices of bread, leaving a little border to reduce sogginess.

Divide the shredded chicken evenly over the avocado layer and pat lightly to level the filling.

Top with a slice of cheese on each sandwich. Close each sandwich with the remaining bread slices and press gently.

Heat a skillet over medium heat and add the butter. Let it foam but do not let it brown.

Place the sandwiches in the skillet. Cook 3 to 4 minutes per side, pressing lightly with a spatula, until the bread is golden and the cheese is melted.

Remove from skillet, let rest for a minute, then cut and serve warm.