- 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!