Cover the cashews with boiling water and let them soak for 15 minutes while you start the soup, then drain.
Heat the olive oil in a large pot over medium heat and cook the onion for 5 minutes until soft and translucent. Add the garlic and cook 1 minute more.
Add the potatoes, the diced carrot, and the vegetable broth. Bring to a boil, then simmer for 15 minutes until the potatoes are fork tender.
Scoop out about 3 cups of the potato and carrot mixture with some broth and add it to a blender along with the drained cashews, nutritional yeast, dijon, smoked paprika, turmeric, salt, and pepper. Blend on high for 2 minutes until completely smooth and glossy.
Meanwhile, add the broccoli florets and the shredded carrot to the pot still on the stove and simmer for 6 to 7 minutes until the broccoli is tender but still bright green.
Pour the blended cashew mixture back into the pot along with the oat milk. Stir well and warm through over low heat without letting it boil.
Finish with the lemon juice and taste for salt. The acid is what makes it read as sharp cheddar, so do not skip it. Serve in bowls with crusty bread.