Heat oven to 375° F.
Butter a 8"x8" baking dish with half of the butter.
Place 4 slices of the bread in the bottom of the baking dish in one layer.
Heat a frying pan over med-high heat. When it is hot, add the remaining 1 Tbsp of butter and the olive oil. Then add the onion and cook, stirring occasionally, until there is some slight browning on the onion, and it is soft and translucent.
Add the spinach, dill, salt, and pepper to the frying pan and mix well.
Spread the spinach mixture evenly over the slices of bread in the baking dish.
Sprinkle on 1 cup of the shredded Swiss cheese.
Arrange the remaining 4 bread slices over in one layer.
In a 4 cup measuring glass or bowl, beat the eggs, milk, and 1/4 tsp salt together well.
Pour egg mixture over the bread evenly.
Sprinkle on the remaining Swiss cheese and the parmesan cheese.
Let sit in the fridge for 1 hour or longer before baking, if possible.
Bake, uncovered for 35-45 minutes or until the strata is puffed and golden in color. If it is browning too quickly, cover loosely with foil that has been sprayed with cooking spray.
Remove to a wire rack. Let stand 10 minutes before serving.