In a medium bowl, whisk together the eggs, whole milk, salt, and black pepper until fully combined.
Heat the unsalted butter in a non-stick skillet over medium heat until melted and foamy.
Pour the egg mixture into the skillet and cook, stirring gently with a spatula, until the eggs are softly scrambled and just set, about 3-4 minutes. Remove from heat.
Preheat a panini press or grill pan over medium heat.
Brush one side of each slice of ciabatta bread with olive oil.
Place two slices of bread, oiled side down, on a clean surface. Divide the scrambled eggs evenly between the two slices.
Top the eggs with fresh spinach leaves and two slices of cheddar cheese per sandwich.
Cover each sandwich with the remaining slices of bread, oiled side up.
Place the sandwiches on the preheated panini press and cook for 4-5 minutes, or until the bread is golden brown and crispy and the cheese is melted.
Remove the paninis from the press and let them cool for 1 minute before slicing in half and serving.