Pulse onion, peppers, and celery in food processor until coarse chopped (don't purée). Add oil to Dutch oven and heat over medium heat until shimmering. Add chopped vegetables, season with salt, and cook, stirring occasionally, until vegetables have softened and are beginning to turn golden, 12 to 15 minutes.

While the vegetables are cooking, make a roux: Place flour and butter in microwave-safe measuring cup. Heat on high power until butter is just melted, about 1 minute. Stir to combine and continue to microwave, stirring every minute, until roux is a uniform colour of deep milk chocolate, 8 to 10 minutes.

When vegetables have cooked, stir in garlic, tomato paste, cajun seasoning, smoked paprika, 1/4 teaspoon cayenne, 1/2 teaspoon black pepper, and cook until fragrant, about 1 minute. Stir in roux.

Slowly stir in chicken broth and beer until mixture has thickened. Stir in okra, chicken, 1/2 teaspoon salt, and bring to simmer. Cover and simmer (adjusting heat if necessary) until okra is softened and chicken is tender, about 20 minutes. Season to taste, serve with rice, season with hot sauce, and garnish with green onions.