Cook the Cajun chicken filling: Heat olive oil in a large skillet over medium-high heat. Add the ground chicken and cook, breaking it up with a spatula, until no longer pink and lightly browned. Stir in the onion and cook until softened, then add garlic and Cajun seasoning and cook just until fragrant.
Build the sauce base: Stir in the tomato paste until the chicken is evenly coated. Pour in the chicken broth, tomato sauce, and Worcestershire sauce, then add the brown sugar if using. Bring to a simmer and cook uncovered until thick and spoonable, stirring occasionally. Season with salt and pepper to taste.
Make the cheese sauce: In a small saucepan, melt the butter over medium heat. Whisk in the flour and cook for about 1 minute. Slowly whisk in the milk and cook, whisking often, until slightly thickened. Turn heat to low and stir in the cheddar cheese and paprika until melted and smooth. Season with a pinch of salt.
Toast the buns: Spread the cut sides of the brioche buns with softened butter and toast in a dry skillet or under the broiler until golden and crisp at the edges.
Assemble and serve: Spoon a generous amount of the Cajun chicken filling onto each toasted bun. Ladle the warm cheese sauce over the top, garnish with green onion or parsley if using, cap with the top bun, and serve immediately.