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Cooking roast on the stove top allows the roast to absorb the flavored liquid. Excess fat prevents the meat from completely absorbing all of the flavoring. After trimming the fat from the roast, season the entire roast with pepper. The Dutch oven should generally be able to hold five quarts. Heat it to a medium-high temperature. Once the meat has browned, add 2 1/2 cups (567ml) of water, two beef bouillon cubes and a bay leaf to the Dutch oven until it boils. When the pot reaches a boil, cover the Dutch oven with a lid and simmer the eye round roast for 50 minutes over a low flame. While it simmers, the roast will become tender and infused with flavor. After 20 minutes, add chopped carrots, potatoes, celery, etc. to the pot. Heat the liquid to boiling and once it boils, reduce the heat and let it simmer for 30 more minutes. If you have a meat thermometer, take the roast off when it reads 135ºF (57.2ºC). If you do not have a meat thermometer, cut into the meat. If the inside is still red, keep cooking it until it is a nice pink color. Let the meat marinate in its juices, then slice the roast into thin portions.
Remove excess fat from the surface of the roast. Place a large Dutch oven on the stove. Place two teaspoons of oil (olive or salad) into the pot and add the roast. Add the other ingredients to the pot. If you would like to boil carrots and potatoes in the mixture with the roast, let the mixture and the roast simmer for 20 minutes. Remove the roast from the pot when it is done. Allow the eye round roast to sit for 15 minutes.