Heat oil in 12-inch skillet over medium heat until shimmering. Add onion and ½ teaspoon salt, cover, and cook, stirring occasionally, until onion is soft, about 5 minutes. Add Cajun seasoning and cayenne and cook until fragrant, about 1 minute. Add meat and cook, breaking up pieces with spoon, until just pink, about 3 minutes.
Stir in tomato puree, ketchup, and ¼ teaspoon pepper. Simmer until sauce is slightly thicker than ketchup, 8 to 10 minutes. Season with salt and pepper to taste. Spoon meat onto hamburger buns and serve.