Add the frozen broccoli, celery, onion, garlic powder, nutmeg, pepper, cream cheese, chicken broth, chicken base, evaporated milk, and flour straight into your slow cooker. Give everything a good stir so the cream cheese starts to break up a bit and the flour doesn't clump. You don't need to pre-cook anything at this stage.
Pop the lid on and set your slow cooker to low for 3 to 4 hours. You'll know it's ready when the broccoli is fork-tender and the whole kitchen smells amazing. I usually check around the 3-hour mark since slow cookers can run a little differently from one another.
Once the broccoli is tender, add the Parmesan and shredded cheddar cheese. Stir until everything is fully melted and smooth. This is the step that turns your broth into that classic thick, cheesy soup texture everyone loves.
Turn off the slow cooker and let the soup sit, covered, for about 8 to 10 minutes. This resting time lets it thicken up a bit more and lets the cheese finish melting evenly. Taste and adjust the salt and pepper before ladling it into bowls.