Heat the olive oil in a large pot over medium heat. Add the onions, celery, and carrots to the pot. Season with salt and pepper. Cook, stirring occasionally until vegetables start to soften, about 5 minutes.
Stir in the chopped potatoes, thyme, and garlic and cook until fragrant, about 1 minute. Add the vegetable broth. Bring mixture to a boil, then simmer until the vegetables soften, about 20 minutes.
Use an immersion blender to lightly blend the soup so it’s thicker but still chunky. You can remove some of the vegetables aside to stir back in if you prefer to have more whole vegetables.
Stir in the peas, corn, and milk and simmer until the vegetables are warmed through and soup has thickened, about 5 minutes.
Serve with fresh parsley and crusty bread, if desired.
Video
Notes
Storage: Store any leftovers in an airtight container. They will last about 4 days in the fridge and reheat really well.