Heat 6 cups water in a large pot, and when it is boiling, add greens. Stir to submerge, and boil gently for 20-25 minutes. Drain in colander and set aside when done.
While greens are boiling, heat oil in a 12-inch non-stick skillet and add chicken and chopped onion. Saute, stirring occasionally, over medium-high heat for about 10 minutes or until chicken is done. Turn heat down to medium if it begins to burn. Remove chicken to a bowl and set aside.
In the same pan (do not clean it), place mole paste and 1 cup water. (Use a knife to slice through mole paste to make it easier to remove from jar.) Heat and stir, mashing paste with a fork or the back of a spoon, until the paste begins to combine with the water. Add another cup of water and continue mashing and stirring. As the mole combines with the water and heats up, it will begin to thicken. Add the final cup of water and brown sugar when the paste is nearly all combined. (If you buy a different brand of mole than recommended, it might be more or less sweet, so add sugar carefully, tasting as you go. You dont want to create too sweet of a dish, since it is a main course.) Once it is smooth, continue cooking on a gentle simmer and stirring for another minute or two to blend flavors. The whole mole process will take about 10 minutes.
By this time, your greens will be done and drained. Return drained greens to the empty pot in which you cooked them, add cooked chicken-onion mixture, and 1 to 1 1/3 cups of the prepared mole sauce. (Save rest of mole sauce for another use. It will freeze well.) Stir to combine, heat through, and serve over hot cooked rice.
Note: You may use any favorite mole paste, but Dona Maria brand in a small jar is commonly available in grocery stores, usually sold in the aisle with other Hispanic foods such as chiles and salsas. There is also a mole version from Dona Maria called Adobo, which seems even hotter, so if you crave spicy things, you might like to try that version.
What to do with leftover mole sauce? Serve it as a sauce over roasted or grilled chicken or pork.