Naturally Dyed Pickled Easter Eggs

Name: Naturally Dyed Pickled Easter Eggs
Ingredients: 6 hard-boiled eggs, peeled; 1 cup distilled white vinegar; 2 teaspoons kosher salt; 2 teaspoons sugar; 1/2 small beet, peeled, quartered; 1 shallot, sliced; 1 bay leaf; 1 (1/2-inch) piece ginger, thinly sliced; 2 teaspoons black peppercorns; 1/4 teaspoon turmeric powder; 1 cup chopped purple cabbage; 1 teaspoon caraway seeds; 2 teaspoons baking soda, divided; 1 medium carrot, peeled, sliced; 2 garlic cloves; 1 sliver peeled beet; Small pinch of saffron