Cook the pasta: Bring a large pot of heavily salted water to a boil over high heat. Add the penne and cook until al dente, which is typically 1-2 minutes less than the package instructions. Reserve 1 cup of pasta water — if you're planning to reheat leftovers later, save an extra cup in the fridge. Drain the pasta and set aside.
Cook the chicken: Heat 2 tablespoons olive oil in a large skillet over medium heat. Season the sliced chicken with kosher salt. Cook, stirring occasionally, until fully cooked and lightly golden. Remove from the skillet and set aside.
Make the sauce: In the same skillet, add the remaining olive oil and butter. Let the butter melt completely, then whisk them together so the fats combine. Add the garlic and red pepper flakes (if using) and cook until fragrant, about 30 seconds. Stir in the tomato paste and cook, stirring constantly, until it turns a deep red color, about 3 minutes.
Incorporate the cream and vodka: Reduce the heat and add the heavy cream, whisking constantly. It will take a little time for the cream to fully combine with the tomato paste, so keep whisking until the sauce is smooth, about 5 minutes. Pour in the vodka and continue whisking, cooking for another 5 minutes until the alcohol burns off and the sauce is fully incorporated.
Thicken the sauce: Stir in 1 teaspoon kosher salt and ½ cup of the reserved pasta water. Let the sauce simmer, whisking occasionally, until it thickens enough to coat the back of a spoon without sliding off, about 20 minutes.
Combine the ingredients: Add the cooked pasta to the sauce first, tossing to coat evenly. Then add the cooked chicken, stirring everything together. If the sauce is too thick, add more reserved pasta water as needed.
Finish and serve: Sprinkle with grated Parmesan or pecorino cheese and serve warm.