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In a large skillet, heat the butter and oil over medium-high heat. Add the onion, celery, bell pepper (AKA “The Trinity”), and garlic and saute until they are soft and tender, about 8 minutes.
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Stir in the flour until it is heated through, about 2 to 3 minutes. You want your roux to get some color. Gradually pour in the broth in 1/4 cup increments, stirring after each addition, until the mixture has thickened.
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Add the tomatoes, thyme, cayenne, salt, paprika, oregano, and hot sauce. Reduce heat to medium, cover, and simmer the mixture for about 10 minutes, stirring occasionally.
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Stir in the shrimp and parsley and cook the etouffée until the shrimp is no longer pink, 3 to 5 minutes. Serve over hot rice.