Adjust oven rack to lower middle position and heat to 300 degrees.
Combine all ingredients in a large dutch oven, including the spent orange halves and juice.
Bring the mixture to a boil over medium-high heat, uncovered. Once it simmers, cover pot and transfer it to the oven.
Cook until the meat falls apart when prodded with a fork, about 2 hours.
Remove the pot from the oven and use a slotted spoon to remove the meat from the pan and place it on a large foil-lined jelly roll pan.
Remove and discard everything from the pot except the cooking liquid.
Place pot over medium-high hear on the stove and boil until thick and syrupy, about 20-25 minutes. You should have about 1 cup of liquid remaining when it is finished.
While the liquid is reducing, break each cube of pork into three pieces. Once the liquid has become syrup, gently fold the pieces of pork into the pot, trying not to break up the pork any further.
Spread the pork back onto the foil-lined pan and evenly spread so there is a single layer. Place the pan of meat on an upper middle rack of the oven and broil on high until the top of the meat is well browned and the edges are slightly crisp, about 7-8 minutes. Using a wide spatula, flip the pieces of meat the broil the other side the same way.
Serve in a tortilla will your favorite toppings.