Place 4 large boneless, skinless chicken breasts on cutting board. Sprinkle 1 tablespoon Cajun seasoning over chicken breasts on both sides, adding more seasoning to taste.
Place large skillet on stovetop over medium-high heat. When skillet is warm, add 1 tablespoon neutral oil and tilt or swirl pan to distribute oil across entire surface. Continue heating skillet until oil is hot and shimmery.
When oil is ready, place seasoned chicken breasts in skillet. Sear chicken 3 to 5 minutes or until golden.
Flip chicken over and sear 3 to 5 minutes on other side, or until chicken reaches internal temperature of 162° Fahrenheit. Once chicken reaches target internal temperature, transfer chicken to plate and set aside.
Return skillet to stovetop and reduce heat to medium, then add 2 tablespoons unsalted butter. Let butter melt, swirling and tilting pan occasionally to distribute butter across surface.
Once butter has melted, add 1 cup holy trinity. Stir to incorporate, then sauté onions, celery, and bell pepper 5 minutes or until vegetables are softened and onions are translucent.
Add 4 cloves garlic and stir to incorporate. Sauté garlic 30 to 60 seconds or until fragrant. Once garlic is fragrant, add 1 tablespoon Cajun seasoning and ½ teaspoon freshly cracked black pepper and stir to incorporate.
Pour ¼ cup dry white wine into skillet and stir, scraping bottom of pan to loosen any browned bits of food that may have stuck to surface. Once pan is deglazed, raise heat to medium-high and bring liquid to boil.
Once liquid begins to boil, immediately reduce heat to medium-low. Let mixture simmer 2 to 3 minutes or until liquid has reduced by approximately 50%.
Add ½ cup low-sodium chicken stock and stir to incorporate, then let mixture simmer until liquid has reduced by approximately 25%.
Reduce heat under skillet to low. Remove skillet from heat and let cool slightly, then slowly pour in 1 cup heavy cream (at room temperature). Stir to incorporate cream, then return skillet to stovetop over low heat.
Add salt to taste and stir to incorporate, then return chicken breasts to skillet. Spoon sauce over tops of chicken breasts.
Transfer chicken to serving plates with mashed potatoes, cauliflower, rice, pasta, or zoodles. Spoon remaining sauce over chicken and serve immediately.