Heat the oil in a Dutch oven or large stockpot over medium heat. Add the onions, celery, and carrots. Cook until the onions and celery are soft, about 7-10 minutes. Reduce the heat and add the garlic. Cook for 1 minute, stirring constantly.
Add the chicken broth and cauliflower. Bring to a low boil and cook until the cauliflower is fork tender, about 15 minutes.
Meanwhile, in a small skillet over medium heat, cook the bacon until crispy. Add the butter to the skillet with the bacon and any residual bacon fat. Once melted, add the bread crumbs and continue cooking until browned, stirring very frequently. Set aside.
Remove 1 cup of hot broth from the soup. Whisk together with the flour. Whisk it into the soup and continue cooking until it has slightly thickened. Reduce the heat to low. Whisk in the shredded cheese 1 cup at a time stirring to melt. If a thicker consistency is desired, use an immersion blender or stand blender to puree a portion of the soup. Be sure to remove the center cap on the stand blender for the steam to release.
Stir in the Cajun seasoning. Season to taste with salt and pepper. Remove from the heat. Stir in the cream and top with fresh or dried parsley or chives. Spoon into bowls and top with 1-2 tablespoons bacon breadcrumb mixture.