Hard boil the eggs (15 minutes in simmering water). Using a slotted spoon, transfer the eggs from the hot water into a bowl of ice water and allow to cool for a few minutes. Peel eggs and slice carefully in half the long way. Set the whites aside in a tray in the refrigerator. Collect the egg yolks into a small bowl. Add mayonnaise, mustard, 1 tsp vinegar and a pinch of salt to the yolks and mash with a fork until well mixed and smooth in texture. Refrigerate this mixture while you prepare the dyes.
Peel and dice the beets. Place in a pot with 1 1/2 cups of water. Bring to a boil, simmer for 20 minutes. Chop purple cabbage and place in a pot with 2 cups of water. Bring to a boil, simmer for 20 minutes. Put 1 1/2 cups of water in a pot and stir in 2 tsp turmeric. Bring to a boil, simmer for 10 minutes.
Turn off all burners. Drain beet water into a bowl. (These cooked beets can be used for salads or other recipes.) Stir in 1 Tbsp vinegar and 1/2 tsp salt. Pour turmeric water into a bowl and stir in 1 Tbsp vinegar and 1/2 tsp salt. Drain cabbage water into a bowl. Stir in 1/4 tsp baking soda and 1/2 tsp salt. Be sure the kids are watching when you add the baking soda! Before adding it, the water will look purple and after adding, it will turn blue!! Allow all dyes to cool. Drop egg whites into bowls of dye and allow dye to permeate eggs. The beets will take the least time and you may pull some out early if you’re looking for a lighter pink. When the egg whites are colored, pull them out, rinse briefly under water and lay on paper towels or clean rags to dry.
You can fill all the colored whites with the yellow yolk mixture, or if you like, take a tiny amount of the beet red dye and mix it in with some of the yolk mixture to make some of the filling pink. Fill your deviled easter eggs and serve on a platter!