Prep Work: Drain and reserve the juice from the diced tomatoes. You should have about ¼ cup of juice, which will be used to deglaze the skillet after searing the chicken. Combine all the sauce ingredients in a large measuring cup with a spout and set aside.
Prepare the Chicken: Slice each chicken breast in half lengthwise to create two thinner slices. Cover with plastic wrap and use the textured side of a meat mallet to pound the chicken to ½ inch thickness. Season each side with Cajun seasoning, then sprinkle with flour and rub it over the surface of each side.
Heat oil in a large skillet over medium-high heat. Sear the chicken in batches for 4-5 minutes per side, until a golden crust forms. Set the seared chicken aside.
Make the Sauce: Add the reserved tomato juice to the skillet and reduce the heat to medium. Use a silicone spatula to scrape the bottom and sides of the skillet, incorporating the browned bits (fond) for added flavor.
Add the butter and minced garlic to the skillet, cooking for 1 minute until fragrant. Stir in the flour and cook for 2 minutes to eliminate the raw flour taste.
Gradually add the sauce mixture to the skillet, stirring continuously to combine. Bring the sauce to a boil, then reduce to a gentle simmer. Add the drained tomatoes and let the sauce reduce and thicken for 5-7 minutes.
Lower the heat to low and slowly sprinkle in the shredded cheese, stirring until fully melted and incorporated into the sauce. Stir in the lime juice.
Return the chicken to the skillet, spooning the sauce over the top. Let it simmer uncovered for 4 minutes, allowing the flavors to meld. Add fresh lime wedges to the skillet and simmer for 1 additional minute.
Serve: Remove the skillet from the heat. Sprinkle with freshly chopped cilantro and serve the Cajun Chicken with rice, beans, or a fresh salad.