Place milk in a small saucepan, and heat until a candy thermometer registers 110 degrees. Pour milk into the bowl of an electric mixer, and whisk in 1 1/2 cups all-purpose flour, the sugar, and the yeast. Cover, and let stand until bubbling, about 30 minutes.
Attach bowl to mixer fitted with dough hook. On medium-low speed, add whole eggs, one at a time. Add remaining cup all-purpose flour and the salt. When incorporated, add butter, orange zest, orange juice, and vanilla. Gradually add enough bread flour to form a soft, slightly sticky dough. Generously butter a large bowl, and place dough in greased bowl; cover tightly with plastic wrap. Set aside in a warm place to rise until 1 1/2 times its original size, about 1 1/2 hours.
Heat oven to 400 degrees. Line an 11-by-17-inch baking pan with parchment paper. Punch down the dough. If making two loaves, divide dough in half, and divide each half into three equal portions. Roll out each portion on a lightly floured surface into a 16-inch-long rope. Loosely braid ropes together, pinching ends to seal; place on pan. Place 4 red eggs down center of each braid, heavy ends down. If making a ring, divide dough into thirds. Roll out each piece into a 30- inch-long rope. Loosely braid together, and join two ends together, forming a ring; place on pan. Place eggs, heavy ends down, around ring. Cover with greased plastic; set aside to rise until doubled in size, about 1 1/2 hours.
Bake 15 minutes. In a small bowl, whisk egg yolk with 1 tablespoon water; brush bread with egg wash. Reduce heat to 350 degrees; continue baking until brown and hollow when tapped on bottom, about 25 minutes more. If bread becomes too dark, cover with aluminum foil. Cool on a wire rack; serve.