To a large dutch oven, heat the olive oil over medium heat. Add the onion and season with salt and black pepper. Cook for 5-6 minutes or until tender. Add the garlic and cook for another minute, or until fragrant. Add the beer. Bring to a boil and then lower to a simmer. Simmer for about 15 minutes or until the beer is substantially reduced.
Add the chicken broth and taking an immersion blender, blend the mixture until completely smooth. Alternatively, you can use a regular blender if you don't have an immersion blender on hand.
Bring the soup to a boil and then lower to a simmer. Simmer for an additional 15 minutes.
In a large bowl, whisk together the flour and just enough milk to create a paste (about 1/4-1/3 of a cup). Whisk the flour into the soup and then add the remaining milk. Continue to simmer the soup for another 15 minutes, whisking frequently until the soup has thickened slightly. Stir in the hot sauce. Cook for an additional 3 minutes and then remove from heat.
In 1/2 cup increments add the cheddar cheese and blue cheese crumbles to the soup, stirring and melting the cheeses completely before adding the next addition. It's ok if a few chunks of blue cheese remain.
Stir in the shredded chicken breast.
Serve the soup hot, garnished with croutons. chives, bacon, blue cheese and a drizzle of hot sauce. Enjoy!