Place a large soup pot or dutch oven over medium-high heat and melt butter. Add onion and cook 2–3 minutes until onion begins to soften. Add white wine and cook until the wine is almost completely evaporated.

Sprinkle onions with flour then use a whisk to stir for about 1 minute. Add chicken broth while whisking constantly and continue whisking until flour is completely incorporated. Add broccoli, half-and-half, salt, pepper and garlic powder and stir to combine.

Bring mixture to a boil stirring frequently, then reduce heat to medium-low and simmer for 20 minutes.

Add Mexican cheese blend to the soup a handful at a time, stirring until melted before adding the next handful. Add sharp cheddar cheese, stir until melted then remove from heat. Season to taste with additional salt and pepper and thin the consistency by adding a bit more broth if desired.