When you take these eggs out of the pickling liquid, they will be a blue-lavender color, but will turn teal green as they sit out and oxidize.
For the pickling:1.
In a small saucepan, combine vinegar, sugar, salt, pepper, caraway seeds, and 1/2 cup water. Bring to a simmer and let cook for 3 minutes, stirring occasionally to dissolve the sugar. Add cabbage and baking soda and simmer until the color turns blue, adding more baking soda if needed to get the color you like.2.
Using a slotted spoon, place a third of the cabbage mixture in a 1 quart jar or container. Top with 3 eggs. Add another third of the cabbage, then remaining eggs. Top with remaining cabbage. Cover with pickling liquid, adding water if needed to cover the top layer of cabbage. Seal jar and gently swirl the liquid around.3.
Let eggs pickle at room temperature for 4 hours. At this point you can either use them, or refrigerate for up to 1 week. The longer they sit in the pickling liquid, the stronger in flavor they become.
For the deviled eggs:
Remove the eggs from the jar and cut them in half. Scoop yolks into a bowl and use a fork to mash them. Add the mayonnaise, sour cream, lemon juice, and salt, to taste. Either spoon the yolk mixture into the whites, or place yolk mixture in a resealable plastic bag with a tip cut off, and squeeze into the whites. Garnish with shallot and some of the blue cabbage if you like.