In a large pot over medium heat, drizzle olive oil and add finely chopped onion and minced garlic. Sauté for about 3–5 minutes until the onions become translucent and fragrant.

Stir in diced potatoes and continue cooking for about 5 minutes, stirring occasionally until the potatoes start to soften.

Carefully pour in the vegetable broth, season with salt and pepper, and bring to a boil. Reduce heat and let simmer for about 10 minutes until the potatoes are fork-tender.

Add fresh broccoli florets to the pot and cook for an additional 5 minutes until the broccoli is tender but still vibrant.

Use an immersion blender to purée half of the soup, blending until smooth while maintaining some chunky pieces.

Return the blended mixture to low heat and stir in the shredded sharp cheddar cheese until melted. Then, pour in the heavy cream or milk.

Ladle the creamy soup into bowls and garnish with additional cheese, fresh herbs, or a sprinkle of pepper.