Stir together the sponge ingredients in a 1 quart sized bowl. Cover with plastic wrap and allow to ferment at room temperature for 1 hour. The sponge will foam and collapse on itself when you tap the bowl. If it doesn't foam you will need new yeast and repeat the process.
While the sponge is fermenting, grate your cheese and chop your caramelized onion.
In the bowl of an electric stand mixer, fit with the paddle attachment, add flour, salt, and sugar. Mix or whisk to combine. Add eggs and sponge and mix together on low speed until the ingredients form a coarse ball. If you have loose flour in the bottom drizzle a small amount of water or milt to gather it into the ball. Mix for 1 minute and then let the dough rest for 10 minutes to allow the gluten to develop.
Divide the butter into 4 pieces. On medium speed, mix the butter into the dough one piece at a time. The dough will be soft but not like a batter. Continue mixing until the dough is a smooth tacky mass, about 8 minutes. Scrape the sides of the bowl down occasionally if necessary. Replace paddle attachment with a dough hook attachment and continue mixing on a medium speed for 4 minutes or until dough comes off the sides of the bowl. If it doesn't do this add a small amount of flour.
Add the cheese and onions and mix on low until just combined. If the dough is sticky, add another small amount of flour until dough is tacky but not sticky. Transfer dough to a large well oiled bowl, rolling to coat all sides with oil. Cover with plastic wrap and let rise until dough increases at least 1-1/2 times, 90 minutes.
Leave dough in one piece for a large loaf or divide in half for two smaller loaves. Place dough in an 8-inch round cake pan or 2-8" loaf pans that have been sprayed with non-stick cooking spray. Spray the top of the dough with cooking spray and cover loosely with plastic wrap or a clean towel. Let rise until dough just reaches the top of the pan(s), 60-90 minutes.
With the oven rack set in the lower 1/3 of the oven, preheat to 350°F. Bake for 20 minutes and then turn the pan. Bake for an additional 20 to 30 minutes or until bread reaches an internal temperature of 185°-190°F.
Remove the bread from the pan(s) onto a cooling rack. Allow the bread to cool for one hour before serving.
Recipe adapted from the book "The Bread Baker's Apprentice" by Peter Reinhart *Prep time includes rising time. **For the dough, I recommend using an electric stand mixer. This dough is tacky and somewhat difficult to knead by hand.
This recipe was calculated using the exact brands and measurements I used to make this recipe. If you are following a strict diet please note changing anything will cause the nutritional info to change. My calculations are intended as a guide only.