Cheesy Chorizo Swirl Bread

Prepare the Dough: Place 2 1/2 cups of the flour in the bowl of an electric mixer fitted with a dough hook. Add warm milk, warm water, olive oil, salt, beaten egg, and yeast; beat on medium-low speed, gradually adding up to 1/2 cup more flour as needed, until dough begins to pull away from sides of bowl, about 4 minutes. Increase speed to medium, and beat until dough is smooth and elastic, about 2 minutes. (Dough will be sticky.) Transfer to a lightly greased bowl; turn to grease top, and cover loosely with a clean towel. Let rise in a warm place (80°F to 85°F), free from drafts, until doubled in bulk, about 1 hour.

Meanwhile, prepare the Filling: Cook chorizo in a large skillet over medium-high, stirring often, to break into pieces, until browned, about 6 minutes. Using a slotted spoon, transfer chorizo to a plate lined with paper towels to drain. Add oil and onions to chorizo drippings; reduce heat to medium, and cook, stirring occasionally, until very tender and translucent, about 20 minutes. Set aside.

Punch dough down, and turn dough out on a lightly floured surface. Roll into a 22- x 9-inch rectangle. Spread chorizo mixture over dough, and top with onions, cheese, oregano, and garlic. Roll up dough, jelly-roll fashion, starting at 1 long side. Pinch seam to seal, and place, seam side down, on a parchment paper-lined baking sheet. Carefully make a 1-inch-deep cut down center of dough, starting about 1 inch from end, and ending 1 inch from opposite end. Form dough into an "S" shape, tucking both ends under the center and pinching together to seal. Cover with clean towel, and let rise in a warm place until doubled in size, 45 minutes to 1 hour.

Preheat oven to 350°F. Whisk together 1 egg and 2 teaspoons water in a small bowl. Brush dough lightly with egg mixture.

Bake in preheated oven until golden brown, 35 to 40 minutes, tenting with foil after 30 minutes if needed. Transfer bread to a wire rack, and cool 15 minutes. Serve warm or at room temperature.