Heat a heavy-bottomed pot over medium heat and add the olive oil. Once it shimmers and begins to sizzle, add the diced onion. Sauté until translucent and fragrant, about 5 minutes, until the onion is soft and slightly golden.
Add the minced garlic to the pot and cook for about 30 seconds, stirring constantly until aromatic. Be careful not to burn it.
Pour in the vegetable broth and bring to a gentle simmer. Then, add the broccoli florets, cover the pot, and steam for 8-10 minutes until the broccoli is tender but still vibrant green.
While the broccoli cooks, whisk together the plant-based milk, nutritional yeast, lemon juice, and smoked paprika in a small bowl. Set aside.
Once the broccoli is tender, carefully transfer the mixture to an immersion blender or a regular blender. Blend until smooth and creamy, about 30 seconds, being cautious with hot liquids.
Pour the cheesy mixture back into the pot with the blended broccoli. Warm over low heat, stirring constantly, until heated through and slightly thickened, about 3-5 minutes. Add salt and pepper to taste, adjusting seasoning as needed.
Remove from heat and let the soup rest for a couple of minutes to allow flavors to meld. Taste again and add a squeeze of lemon or additional salt if desired.
Serve hot, garnished with a drizzle of olive oil or a sprinkle of nutritional yeast if you like. Enjoy your cozy, vibrant bowl of vegan broccoli cheddar soup!