In a large pot, cover the mandarinquats with cold water by 1 inch and bring to boil that cannot be stirred down.
Immediately reduce heat to low and simmer until the mandarinquats can easily be pierced with a wooden chopstick, about 30 minutes. Add more water if needed.
Drain the mandarinquats, discarding their cooking liquid. Set them aside in a separate bowl.
Combine the vinegar, sugar, ground ginger, pink peppercorns, and salt in your preserving pot and begin to bring to a boil over high heat. (You can use the same one that you just simmered the fruit in.)
Stir to dissolve the sugar and salt. Once they’re dissolved, add the fruit and bring to a boil. Lower the heat and simmer for 5 minutes.
Ladle the kumquats into the prepared half-pint jars, evenly distributing them between the 2 jars and leaving ½ -inch headspace. Remove air bubbles and wipe rims. Place the lids on the jars and screw on the bands until they are fingertip tight.
Process in a water-bath canner for 10 minutes, adjusting for altitude if needed.
After 24 hours, check the seals. Label, date, and store out of direct sunlight without the bands for up to a year.
After you are done eating all the kumquats, you can use the remaining syrup as a drinking shrub to mixed into cocktails!