Remove pits and stems from 2 pounds of cherries and place in a small sauce pan (you may use frozen cherries for this step). Add water and sprinkle with sugar. Bring the cherries to a slow boil. Reduce heat and simmer for 20 minutes.
Place a fine sieve over a mixing bowl. Scoop the cherries into the sieve a little at a time and press the cherries with the back of a wooden spoon to release the juice and pulp into the bowl. Scrape the skins out of the sieve and repeat until all of the cherries have been juiced.
Return the juice to the sauce pan over medium heat and stir in the lemon juice. Combine the bottled cherry juice with the cornstarch to make a slurry and stir until there are no lumps. Slowly pour the slurry into the sauce pan while whisking.
Gently simmer until the cherry juice reaches the desired thickness. It will become thicker as it cools.
Just before removing the sauce from the heat, add the remaining one pound of pitted cherries and fold into the sauce. Allow the sauce to return to a boil and immediately remove from the heat. The cherries should retain a firm texture.
Let the sauce cool and pour into a jar. Keep refrigerated for up to one week.