In the bowl of a stand mixer, add 1 3/4 cups plus 2 tablespoons of the flour, the salt, the lemon zest, and the sugar and whisk to combine.
Make a well in the center of dough and add the milk and the yeast. Use a fork to mix.
Change the stand mixer to the dough hook attachment. Add the 2 slightly beaten eggs to the dough and knead in the mixer until they are just combined, about 1 minute.
Cover the bowl with plastic wrap and let the dough rest, folding the ends into the middle every 30 minutes, until it has risen, about 2 hours.
On medium-high speed, using the dough hook attachment, knead the risen dough and gradually add the butter, until the butter is just combined.
Let the dough rest for 10 minutes.
Knead the dough until it is smooth and does not stick to the sides of the bowl, about 5-6 minutes.
Lighty grease a bowl and add the dough. Cover it with plastic wrap and refrigerate it, about 1-2 hours.
Line a cookie sheet with parchment paper.
Use the extra flour to lightly dust a work surface.
Add the dough to the prepared work surface and divide it into 4 pieces. Roll them each into 10-12-inch-long ropes.
Join 2 of the dough ropes together at the top and twist the ends, one over the other, all the way down. Join the ends to form a braided circle and repeat with the remaining ropes.
Place the braided dough circles on the prepared cookie sheet. Cover and place them in a warm, draft-free area to rest until they rise, about 1-2 hours.
About 15 minutes before baking, preheat the oven to 390 degrees F.
In a small bowl, add the remaining non-beaten egg and the water and beat to combine.
Add 1 dyed, uncooked egg to the center of each of the risen dough circles.
Brush the dough with the egg wash, being careful not to brush the dyed eggs with the egg wash.
Bake the buns until they are golden and make a hollow sound when tapped on the bottom, about 20 minutes.
Immediately transfer the baked buns to a wire rack and let them cool completely.
Serve the baked buns, removing the dyed eggs prior to eating, if desired.