Heat olive oil in a large pot on medium heat. Saute onions until they start to turn translucent, about 5 minutes. Add garlic and heat for another minute will stirring constantly. Stir in flour and cook for another minute while stirring constantly.
Stir in chopped broccoli and potato, making sure to coat all pieces with the sauteed onion, garlic, and flour mixture.
Add chicken broth and bring to a boil. Lower to medium heat and cook for another 20 minutes, or until broccoli and potato pieces soften. Stir in half-and-half.
Transfer everything to a blender and blend until smooth.
Return pureed mix to the pot. Stir in shredded gouda, Serve immediately.