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 hot.
Pour the egg mixture into the skillet and cook, gently stirring with a spatula, until the eggs are softly scrambled and just set, about 3-4 minutes. Remove from heat.
Spread 1 tablespoon of mayonnaise evenly on one side of each slice of sourdough bread.
Place two slices of bread, mayonnaise side down, on a clean surface. Divide the scrambled eggs evenly between the two slices.
Top the eggs with 2 slices of cheddar cheese and 1/4 cup of fresh spinach leaves on each sandwich.
Cover each sandwich with the remaining bread slices, mayonnaise side up, pressing gently to secure.
Preheat a panini press or grill pan over medium heat.
Place the sandwiches on the panini press and cook for 4-5 minutes, or until the bread is golden brown and crispy and the cheese is melted.
If using a grill pan, press the sandwiches gently with a spatula or heavy pan and cook for 3-4 minutes per side until golden and cheese is melted.
Remove the panini from the press or pan, let cool for 1 minute, then slice in half and serve immediately.