Bring a medium pot of generously salted water to boil for the pasta.
In a large pot, heat 1 tablespoon of the olive oil over medium-high heat. Once hot, add chicken and sprinkle with salt and pepper. Cook until chicken is cooked through and use a slotted spoon to transfer the chicken to a bowl. Deglaze the pan with a little water, scraping any cooked-on bits off the bottom of the pan with a wooden spoon. Discard the liquid and wipe out the pot with a paper towel so the bottom of the pot is mostly clean.
Add pasta to pasta water and cook according to box instructions until al dente.
While the pasta cooks, add remaining 3 tablespoons of olive oil and the butter to the pot you used to cook the chicken and heat over medium heat.
Once melted, add garlic and red pepper flakes. Cook for about 1 minute, until garlic is beginning to turn golden and aromatic.
Add tomato paste and cook, stirring often, until paste darkens in color and begins to caramelize on the bottom of the pan, 3 to 5 minutes.
Switch to a heat-resistant whisk and turn heat to low. Carefully add the vodka (the sauce will sizzle and pop aggressively for a few seconds). Whisk, scraping any cooked-on bits from the bottom of the pan as you go, until the vodka is incorporated into the sauce.
Whisk in heavy cream, and heat over low heat, stirring constantly, until sauce is smooth, uniform in color, and heated through. Do not allow the sauce to come to a simmer.
When the pasta is nearly done, use a heat-proof measuring cup to scoop out ¼ cup of the cooking water and whisk it into your sauce. Taste the sauce and add salt to taste.
Once the pasta is al dente, drain or use a slotted spoon or spider skimmer to scoop out pasta, and add it your sauce. Stir to combine.
Stir in cooked chicken. Allow the pasta to sit for 3 to 5 minutes to give the sauce a chance to thicken and the chicken time to warm up in the hot sauce.
Serve and enjoy!