To make the bechamel cheese sauce: Heat a small skillet over medium heat and melt 2 tbsp. butter. Whisk in flour and cook, stirring constantly, for 1-2 minutes. Whisk in milk and seasonings, and stir from time to time until sauce thickens and coats the back of a spoon, about 10-12 minutes.
Add spinach to sauce and allow it to wilt. Remove from heat.
Arrange 4 bread slices on a baking sheet and distribute spinach and sauce evenly over the bread. Top with another slice of bread. Add a small mound of cheese and 2 folded slices of ham on top.
Bake until golden on top, about 10 minutes.
While sandwiches are baking, fry eggs in a skillet using a tablespoon of butter.