Melt butter in a pot over medium heat, then add onions. Cook onions for 3-4 minutes, then add flour. Stir to combine & cook for 1 minute. Pour in skim & evaporated milks. Add nutmeg, broccoli, & a small dash of salt, & plenty of black pepper.
Cover & reduce heat to low. Simmer 20-30 minutes, or until the broccoli is tender. Stir in cheese & allow to melt. Taste & adjust seasonings if needed.
Puree mixture in a blender (dividing mixture in half depending on the size of the blender.) Return to pot over stove, if needed, for re-heating. Serve with a sprinkle of cheese.