Cook the pasta according to package directions. Drain when the pasta is still al dente; do not overcook!
Sprinkle 1 1/2 teaspoons Cajun spice (or more) over the chicken cubes. Toss around to coat. Heat 1 tablespoon butter and 1 tablespoon oil in a heavy skillet over high heat. Add half the chicken in a single layer; do not stir. Allow the chicken to brown on one side, about 1 minute. Flip to the other side and cook an additional minute. Remove with a slotted spoon and place on a clean plate. Repeat with the remaining chicken. Remove the chicken, leaving the skillet on high heat. Add the remaining butter and olive oil. When heated, add the garlic, peppers and onions. Sprinkle on the remaining Cajun spice, and add salt if needed. Cook over high heat for 1 minute, stirring gently and trying to get the vegetables as dark/black as possible. Add the tomatoes and cook for an additional 30 seconds. Remove all the vegetables from the skillet.
With the skillet still over high heat, pour in the chicken broth and wine. Cook for 3 to 5 minutes, scraping the bottom of the pan to deglaze. Reduce the heat to medium-low and pour in the cream, stirring/whisking constantly. Cook the sauce over medium-low heat for a few minutes, until the cream starts to thicken the mixture. Taste and add freshly ground black pepper, cayenne pepper, and/or salt to taste. The sauce should be spicy!
Finally, add the chicken and vegetables to the sauce, making sure to include all the juices that have drained onto the plate. Stir and cook until the mixture is bubbly and hot, 1 to 2 minutes. Add the drained fettuccine and toss to combine.