Heat a heavy cast-iron skillet over medium heat. Add flour and cook, stirring constantly with a wooden spoon, until the flour turns a deep golden color, 7 to 10 minutes. Transfer the flour to a plate and let cool. (There will be a strong aroma similar to burnt toast. Be careful not to let the flour burn; reduce the heat if flour seems to be browning too quickly.) Alternately, toast the flour in a pie plate in a 400°F oven for 20 minutes.
Heat oil in a heavy stockpot over medium heat. Add onion, bell pepper, celery and garlic; sauté until the onions are lightly browned, about 7 minutes. Stir in the toasted flour. Gradually stir in broth and bring to a simmer, stirring. Add tomatoes, okra, pepper, thyme, oregano, cayenne and bay leaf. Cover and cook for 15 minutes. Stir in rice and cook, covered, for 15 minutes longer.
Add shrimp, chicken and sausage; simmer until the shrimp is opaque inside, the chicken is no longer pink and the rice is tender, about 5 minutes longer. Discard the bay leaf and season with salt. Ladle into bowls and serve with hot sauce.
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