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Step 1
Heat 1 tbsp olive oil over medium high heat. Add chicken and andouille and saute until the chicken is no longer pink. Transfer the chicken, sausage, and residual pan juices to a bowl.
Step 2
Reduce the heat to medium, then add the remaining 2 tbsp olive oil and butter; heat until the butter melts. Add the flour and whisk until no lumps remain; continue cooking, stirring occasionally, until the roux is deep golden brown, about 10 minutes. Be patient - the roux will first clump, then eventually smooth out and start to brown.
Step 3
Add the onion, pepper, and celery. Toss to coat in the roux and saute 5 minutes.
Step 4
Add the garlic and saute 1 minute. Add the tomato paste and work into the vegetables; saute until the tomato paste darkens, about 2 minutes.
Step 5
Add ½ c chicken broth to deglaze the pan, scraping up any browned bits from the bottom of the pot. Add the remaining broth, as well as the cajun seasoning and salt. Return the chicken, sausage, and pan juices to the pot.
Step 6
Bring to a low simmer and cook for 15-20 minutes, or until the sauce is thickened and the chicken and sausage are cooked through.
Step 7
Serve over white rice with hot sauce to taste.