Remove the chicken breasts from the package and wash. Set aside.
In a large storage bag, put in one-half cup of flour, a half teaspoon of salt and a half teaspoon of pepper (or to taste). Shake the combination until well mixed. Now, one at a time, add the chicken breasts. Shake them until they're coated with flour. Set each one aside as you've gotten it coated. Toss out any leftover coating mixture. Depending on how big your chicken breasts are, it might even take a bit more flour. That's hard for me to predict.
Heat the vegetable oil on Medium High and add the chicken breasts, breast-side down first. Sauté until that side of the chicken is golden brown. This should take just a few minutes. Turn over and lightly brown the underside of the chicken breast.
Remove the chicken from the sauté pan and put it in an ovenproof dish. Do NOT wash out the sauté; pan! Put the chicken in the oven at 400 degrees to finish the chicken by roasting it. This should take no more than 20 minutes to a half hour.
While the chicken is in the oven, you are going to finish your sauce. Use the same pan that has the drippings in it. With the stove burner turned OFF, deglaze the pan by pouring in the vodka.DO NOT DO THIS WITH THE BURNER ON or you'll risk singing your eyebrows or worse! Use a wooden spoon to scrape up the bits in the bottom of the pan. Add the chicken broth and the lemon juice. You'll want to reduce this mixture by about 25%. Do this on Medium High heat. Then add the tomatoes and the heavy cream. Turn to Medium Low. You are just trying to get the sauce to start to thicken up. Stir often.
When the sauce is finished, turn off the burner if the chicken is not ready yet. You can re-heat the sauce in a few minutes when the chicken is finished. To test your chicken, you can cut a small slice into it; if the juices run clear and you don't see pink, you're ready to go.