Place a large soup pot of Dutch oven oven medium heat.
Drizzle olive oil in the pot and add in the diced red onion. Cook the onions until translucent and then add in the minced garlic. Cook for another 2 minutes.
Add the butter to the Dutch oven and stir it in until fully melted. Sprinkle in the flour and Cajun seasoning. Stir right away so that it absorbs all of the liquids from the pot. Cook the flour mixture for a minute.
Pour the chicken broth slowly into the flour mixture while whisking. Keep whisking the mixture until it becomes smooth, simmers, and thickens.
Stir in the milk and cream into the thickened broth and bring back to a simmer. Add the broccoli florets and chopped bell pepper.
Keep simmering the soup for 10-15 minutes, or until the broccoli is to your desired doneness.
Take the soup off the heat. Add to it ½ cup of the grated sharp cheddar at a time and stir until incorporated. Serve right away.