For the dough, start by stirring together ¼ cup warm water and 2 teaspoons active dry yeast in a medium bowl. Let it stand for about 5 minutes until it becomes frothy.
In a large mixing bowl, combine 3 cups bread flour, ¼ cup sugar, and 2 teaspoons kosher salt.
Using a stand mixer fitted with a paddle attachment, beat ¼ cup softened butter at medium speed until it becomes creamy.
Next, add the frothy yeast mixture and about a quarter of the flour mixture to the stand mixer. Beat until smooth.
Now, add another quarter of the flour mixture, 1 cup milk, and 1 large egg. Beat until smooth.
Gradually add the remaining flour mixture, beating until everything is combined and the dough pulls away from the sides of the bowl.
On a lightly floured surface, turn out the dough and knead for about 10 to 12 times.
Spray a large bowl with cooking spray and place the kneaded dough inside, covering it to rise until doubled in size.
Meanwhile, prepare the filling. Preheat your oven to 400 degrees. Cut about a quarter inch off the top of the head of garlic and wrap it in foil. Bake it until soft, about 1 hour.
In a medium skillet, heat 2 tablespoons olive oil over medium heat. Add the 1 large chopped yellow onion and cook, stirring occasionally, until it turns a rich caramel color.
In the work bowl of a food processor, combine the caramelized onion, roasted garlic paste, ¼ cup softened butter, 2 tablespoons fresh thyme leaves, and ½ cup grated parmesan cheese. Process until the mixture forms a smooth, thick paste.
Once the dough has risen, turn it out on a heavily floured surface and roll it into a 14 by 10 inch rectangle. Spread the onion filling over the dough.
Starting at one long side, roll up the dough jelly-roll style, pressing the edge to seal. Using a sharp knife, cut the roll in half lengthwise.
Carefully twist the two pieces of dough around each other to form a circle. Place the twisted dough into a greased 10 inch cast-iron skillet.
Preheat your oven to 350 degrees. Bake the loaf for 30 minutes. After that, cover it loosely with foil and continue baking until a wooden pick inserted in the center comes out clean.
Once done, take the bread out of the oven and let it cool slightly before serving warm.
