Dredge the chicken breasts in the blackening spice and place in a cast iron skillet over very high heat. Blacken both sides of the chicken and place in the oven for 10 minutes, or until internal temperature of chicken reaches 165 degrees F on an instant-read thermometer. Slice in strips on the bias and set aside.
In a saute pan over medium heat, add 2 tablespoons of extra-virgin olive oil. Add garlic and lightly caramelize. Then add the sun-dried tomatoes and the chicken slices. Deglaze the pan with the white wine. Add the heavy cream, increase the heat to a simmer, and reduce the cream sauce by half.
When the cream sauce is to desired consistency, stir in 1/2 cup Parmesan, salt, pepper, and pasta.
Nest the pasta on large rimmed plates, pour sauce over pasta, and garnish with scallions and the remaining 1/4 cup Parmesan.