In a large Dutch oven or heavy-bottomed pot, melt the butter over medium heat until foaming.

Add the diced onion and sauté for 5 minutes until translucent and soft. Incorporate the minced garlic and cook for 60 seconds until aromatic.

Whisk in the all-purpose flour and cook for 2 minutes, stirring constantly to toast the flour and eliminate the raw taste.

Slowly pour in the chicken broth while whisking continuously to ensure a smooth, lump-free roux base.

Add the chopped broccoli florets, smoked paprika, salt, and black pepper. Bring the mixture to a gentle boil.

Reduce heat to low, cover the pot, and simmer for 10 to 12 minutes until the broccoli reaches a tender, fork-crushable consistency.

Stir in the heavy cream and continue to simmer uncovered for 2 to 3 minutes to allow the liquid to thicken slightly.

Remove the pot from the heat source to prevent the cheese from breaking. Gradually add the shredded smoked gouda in small batches, stirring constantly until the cheese is completely melted and the texture is velvety.

Adjust seasoning with additional salt or pepper if necessary and serve immediately in warmed bowls.