Soft-boil six eggs: Cover them with cold water in a medium saucepan, cook on medium heat until boiling, then cook for 1 1/2 minutes more. Transfer the eggs to a bowl of ice water. Color them, following the dye package's instructions.
In a large bowl, dissolve the yeast in the warm water. Meanwhile, melt the butter in a saucepan, add the milk, and heat until just warm. Pour the mixture into the bowl with the yeast. Add the sugar, two lightly beaten eggs, and the salt. Stir well, making sure the sugar is dissolved and the eggs are incorporated.
Mix in the flour, one cup at a time, until a soft dough forms. Turn the dough onto a floured surface, adding a bit more flour if the dough is too sticky to handle. Knead until the dough becomes elastic, about 1 minute. Place it in a lightly oiled bowl, cover it with a dish towel, and set it in a warm, draft-free area until doubled in size, about 1 hour and 15 minutes.
Punch down the dough. On a lightly floured surface, divide the dough into six equal pieces. Divide each piece again into six equal pieces, for a total of 36. Roll each piece into a ball with a smooth top.
On a lightly buttered cookie sheet, form six dough balls into a circle (about 1 inch of each ball should touch the adjoining ball). Gently press a dyed egg into the circle's center. Repeat to make six loaves on several cookie sheets. Let the loaves rise, covered with dish towels, until they double in size, about 45 minutes.
Preheat the oven to 350°. Beat the remaining egg with 1 teaspoon water and brush the wash over the dough, being sure to coat the seams between the rolls. Bake the loaves for 10 to 15 minutes, or until light golden brown.