In a large saucepan, melt 2 tablespoons of butter over medium heat. Add the diced onion and cook for about 5 minutes, or until tender and translucent, stirring occasionally.

Add the remaining 2 tablespoons of butter to the pan and let it melt. Whisk in the flour, salt, and pepper until a thick paste forms.

Slowly pour in the half and half, water, and chicken broth while whisking constantly. Keep whisking until the mixture is smooth and the flour has fully dissolved.

Stir in the chopped cauliflower. Increase the heat to bring the mixture to a boil, then reduce to a simmer. Cover the pot and cook for about 15 minutes, or until the cauliflower is soft.

Carefully blend the soup using an immersion blender until smooth. You can also transfer it in batches to a countertop blender—just be cautious with the hot liquid.

Return the soup to the heat and bring it back to a light boil. Remove from heat, then stir in the Dijon mustard, cheddar, and pepper jack cheese. Mix until fully melted and creamy.

Ladle into bowls and top with crumbled bacon, extra shredded cheese, or your favorite toppings. Serve hot.