If using a stand mixer: In a large mixer bowl, combine the yeast, salt, eggs, and sugar. Add the warm milk and olive oil and mix. Add 3 ½ cups of the flour and mix with a dough hook (or by hand); adding small amounts of flour until you have a soft, slightly sticky dough. ( You may not need all the flour, just add it until the dough is not wet anymore.) Knead until smooth with either the dough hook or turn out onto a floured board and knead, keeping your hands coated in flour. Place the dough in a lightly oiled bowl, cover and let rise in a warm place until doubled in size, about an hour to an hour and a half.
If using a bread machine: Add the ingredients to the machine in the order listed by the manufacturer, generally, liquids, sugars, flour and finally yeast. Set to dough. Be sure to peek inside while the dough is mixing and add additional flour as needed.
Once the dough has risen, punch it down and turn it onto a floured surface. Divide the dough into 16 even pieces. Roll each piece to form about a ¾ inch thick rope 12 inches long. Take two pieces and braid them together, pinching the ends. Loop into a circle and pinch the ends to close.
Place each braid on a cookie sheet that has been lined with parchment paper (you will need two cookie sheets). Cover and let rise again. ( You can make these to this point a day ahead, cover and keep in the refrigerator removing them an hour before cooking).
Heat the oven to 350°F (325°F if convection). Place the oven racks in the upper and lower third of the oven.
Brush each braid with the egg wash; sprinkle with the sprinkles, if using and place a colored egg in the center of the ring. Bake for 15 minutes for a soft egg, or 20 minutes for a hard egg; switching the cookie sheets halfway through the baking time (not necessary if using convection).
Enjoy your bread with your egg. If soft-cooked, set the egg upright in the "hole" and use a sharp knife to remove the top. If hard-cooked, peel, or cut into half.