Melt the butter in a large pot over medium-high heat.

Add the finely chopped onion and minced garlic. Cook for about 3 minutes, or until the onion is softened.

Mix in the flour, stirring for about 30 seconds to create a roux.

Slowly pour in the milk while stirring continuously, forming a thickish paste.

Incorporate the chicken stock, water, salt, and pepper. Stir well, then add the potatoes to the mixture.

Bring to a simmer, adjusting the heat to prevent rapid boiling. Cook for about 8 minutes, stirring occasionally.

After the potatoes are nearly cooked, stir in the broccoli. Cook for an additional 2 minutes or until the broccoli reaches your preferred tenderness.

Finally, remove the pot from the stove and stir in the cheese until well incorporated and the soup thickens.

Taste and adjust salt if needed.

Ladle the soup into bowls and top with your choice of toppings.