Special equipment: 3 glass pint jars with lids
Bring the vinegar, 3/4 cup water, 3 tablespoons salt, the coriander seeds, mustard seeds, sugar, bay leaf, onions, turmeric and pepper flakes to a boil in a medium saucepan over medium heat, stirring to dissolve the salt and sugar. Reduce the heat, cover and let the brine simmer for 5 minutes.
Meanwhile, pierce each egg all the way through about 6 times with a cake tester or toothpick. Put half the carrots in the bottom of 3 glass pint jars with lids. Add 2 eggs to each jar and then the remaining carrots. Ladle in enough brine to cover the eggs, add 2 more eggs to each jar and ladle in more brine to cover, leaving about 1/4 inch of space at the top. Close the jars, let cool to room temperature, then refrigerate for at least 24 hours before serving. (The eggs will keep, covered by liquid and refrigerated, up to 5 days.)
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