In a large pot, heat 2 tablespoons of olive oil over medium heat. Add 1 medium diced onion and cook for about 5 minutes until softened and translucent. Stir in 3 minced garlic cloves, cooking for an additional minute until fragrant.

Add 4 peeled and diced medium Yukon Gold potatoes to the pot, pouring in 4 cups of vegetable or chicken broth. Bring the mixture to a rolling boil. Reduce the heat to low, cover, and let simmer for approximately 15 minutes until tender.

Stir in 2 cups of fresh broccoli florets. Continue cooking for about 5 minutes until the broccoli turns bright green and becomes tender.

Using a potato masher or immersion blender, gently mash some of the potatoes to create a creamy consistency while leaving some chunky pieces intact.

Lower the heat and stir in 1 cup of milk along with 1 cup of shredded sharp cheddar cheese. Season with 1 teaspoon of salt, ½ teaspoon of black pepper, and ½ teaspoon of paprika. Stir until the cheese melts and the soup is creamy.

Ladle the soup into bowls. Garnish with chopped green onions or chives if desired.