Allow the brine to cool slightly, and then pour it into the jar over the eggs. A warm brine is okay, though I avoid adding it when it is still piping hot. Make sure to get most of the beets in there! If you happen to run out of brine, simply add another splash of your vinegar of choice to top the jar off.
Next, add a tight-fitting lid, and move the jar to the refrigerator right away.
Finally, let the beet pickled eggs sit and infuse for a few days (at minimum). The longer they steep, the deeper the beet color will get – and the more tangy pickled-flavor the eggs will become! Very large eggs can take up to two weeks to become fully seasoned. We usually let them sit in for 5-10 days before digging in.
If some eggs are pressed up against the sides of the jar (leaving un-dyed white spots), or if any are floating up above the brine level, gently tip and shake the jar every couple of days to move things around.