Make the dye. Add the chopped cabbage, water, and salt to a large pot with a lid. Bring to a boil, then cover the pot, and turn the heat down so it doesn't boil over. Cook for 30 minutes. Drain the cabbage, reserving the liquid. Measure how many cups of cabbage liquid you have (it should be about 7), and add it to a large bowl. Stir in 1 tablespoon distilled white vinegar for each 1 cup of liquid. Cool to room temperature, about 1 hour.
Dye the eggs. Add the hard-boiled eggs to the liquid, making sure they’re submerged. Transfer the bowl to the fridge and let the eggs soak until they reach your desired shade (the color gets darker the longer they soak). The eggs pictured here soaked for 4 hours, but you can go as long as 24 hours for more intense color.
Air dry the eggs. Once the eggs reach your desired shade, use tongs to transfer them to a paper towel-lined tray to dry completely before adding brown speckles. Once the eggs are dry on the first side, flip them over and let them dry on the second side.
Add brown speckles. Once the eggs are dry, stir together the vodka and cocoa powder in a small dish. Dip a clean paintbrush (or unused toothbrush) into the liquid and flick it onto the eggs to create a speckled look. Let the speckles dry before serving. (Once dry, you can flip the eggs over and add speckles to the other side if desired.)
Notes
Eggs: Any color of eggs will work; however, use white eggs for the color pictured here.
Cocoa Powder: If you want black speckles instead of brown speckles, use black cocoa powder.
Amount of Dye Made: This recipe makes about 7 cups of dye, which is enough to color 12 hard-boiled eggs at one time in a large bowl (the eggs need to be fully submerged in the liquid). You can reuse this dye for at least one more batch of a dozen eggs after dying the first batch.
Storage: Store hard-boiled eggs in the fridge for up to one week. (Remember to pop the bowl of eggs and dye into the fridge as they soak!)