Rinse and peel onion, potato, and carrot. Rinse broccoli and celery. Peel garlic.
Dice onion and celery. Slice potato and carrots into thin slices. Cut broccoli florets away from the stem. Slice stems thinly. Mince garlic.
Grate cheese.
In a large pot over medium-high heat, heat oil. Add celery and onion. Cook until soft and lightly golden brown, about 6-8 minutes. Add garlic and stir. Cook about 30 seconds more.
Add potato, carrot, broccoli stems, bay leaf, milk, and broth to pot. Bring to a boil. Reduce heat and simmer until veggies are soft, about 15 minutes. Add broccoli florets in the last 10 minutes.
Remove and discard bay leaf. In a blender, puree about half the soup. Return blended soup to pot.
Add cheese, salt, and pepper. Simmer to melt the cheese, 2-3 minutes.
NOTES
Add a dash of ground cayenne pepper or paprika for heat.
Keep pot uncovered when cooking green veggies like broccoli. This will help keep the color.
Be careful when blending hot soup. Fill the blender only half full. Blend in batches, if needed. If your blender lid has a removable cap, remove it. Then, cover the lid completely with a kitchen towel. This allows hot steam to escape.
If you do not have a blender, gently use a potato masher to blend hot soup.