In a medium bowl, whisk together the eggs, whole milk, salt, and black pepper until well combined.
Heat a non-stick skillet over medium heat and add the olive oil.
Pour the egg mixture into the skillet and cook, stirring gently, until the eggs are softly scrambled and just set, about 3-4 minutes. Remove from heat.
Spread the softened butter evenly on one side of each slice of sourdough bread.
Place two slices of bread, buttered side down, on a clean surface.
Divide the scrambled eggs evenly over the two slices of bread.
Top the eggs with two slices of cheddar cheese and half of the fresh spinach leaves on each sandwich.
Cover each sandwich with the remaining slices of bread, buttered side up.
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 with a heavy skillet and cook for 3-4 minutes on each side.
Remove the paninis from the press or pan, slice in half, and serve immediately.