In a small bowl, whisk together the brown sugar, Cajun seasoning, paprika, garlic powder, kosher salt, and black pepper.
Pat the chicken thighs dry with paper towels and evenly sprinkle the seasoning mixture over both sides of the chicken.
Heat a large cast iron skillet over medium-high heat. Add 1 tablespoon of the butter and the olive oil and let the butter melt until shimmering.
Add the seasoned chicken to the hot skillet and sear until browned, about 5–7 minutes per side, until the internal temperature reaches 165°F; transfer the cooked chicken to a plate.
Reduce the heat to medium and add the remaining butter to the skillet. When it melts, add the minced garlic and cook about 30 seconds until fragrant.
Pour in the lime juice and chicken broth, scraping up any browned bits from the bottom of the pan, and simmer the sauce for 1–2 minutes while whisking.
Return the chicken to the skillet and spoon the sauce over the pieces, warming them briefly before serving. Garnish with chopped parsley if desired.