Preheat oven to 375°F. For the blackening seasoning, combine all of the dry ingredients in a medium bowl and set aside.
Heat a large cast iron skillet over highest heat or until scalding hot, about 5 minutes. Add a couple of tablespoons of canola oil. When the oil is very hot, dip the chicken in the melted butter; then thoroughly coat in the blackening rub. Add the chicken to the pan and cook until blackened on each side, about 1-2 minutes per side. Transfer the chicken to a lightly-greased sheet pan. Cook in the oven until chicken reaches an internal temperature of 160°F and the juices run clear.
While the chicken is in the oven, bring a large pot of salted water to a boil. Add the broccoli and cook until bright green and tender, 2-3 minutes. Remove with a strainer to another bowl, season with salt and pepper, and set aside. Return the water to a boil and cook the noodles to al dente according to package directions, about 3-4 minutes. Drain but do not rinse.
To make the sauce, in a heavy saucepan, whisk together the cream, milk, and egg yolk. Add the garlic cloves and bring to barely a simmer, stirring constantly, over medium-low to medium heat. Cook until the sauce is slightly thickened, watching carefully to not cook the egg. Turn off the heat and stir in grated Parmesan. Season with salt and pepper, to taste
Combine the noodles, broccoli, and cream sauce. Season the dish with salt and pepper, to taste. Thinly slice chicken on the bias and place atop the pasta. Garnish with diced tomatoes, bread crumbs, and additional Parmesan cheese. Serve immediately.