Make the dough: In the bowl of a stand mixer, whisk together 2 cups (250g) flour, sugar, yeast, and salt by hand.
In a medium saucepan, heat milk, water, butter, and sour cream over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm milk mixture to flour mixture, and using the paddle attachment, beat at medium speed until combined. Add egg, beating until combined. With mixer on low speed, gradually add 2 cups (250g) flour, beating until combined.
Switch to the dough hook attachment. Beat at low speed until a soft, smooth, and somewhat sticky dough forms and pulls away from bowl, 9 to 10 minutes. Add up to remaining ¼ cup (31 grams) flour if dough is too sticky. Turn out dough onto a lightly floured surface, and shape into a smooth round.
Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.
Spray a 9-inch round cake pan with cooking spray.
Lightly punch down dough. Cover and let stand for 5 minutes. Turn out dough onto a lightly floured surface, and roll into a 21x12-inch rectangle. In a small bowl, combine softened butter and thyme. Spread thyme butter onto dough, leaving a ½-inch border on one long side. Sprinkle and spread Caramelized Onions (see directions below) onto butter. Sprinkle 6 ounces (170g) cheese onto onions.
Starting with long side opposite border, roll dough into a log, pinching seam to seal. Place log seam side down, and cut in half lengthwise, leaving 1½ inches at one end. Turn halves cut sides up, and carefully twist dough pieces around each other; form into a circle. Place, cut sides up, in prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 30 minutes.
Position one oven rack in center and one at top of oven. Preheat oven to 350°F (180°C).
Bake on center rack for 30 minutes. Sprinkle remaining 2 ounces (57g) cheese on top, and bake until cheese is melted and golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 15 to 20 minutes more, placing a piece of foil on top rack of oven to prevent excess browning, if necessary. Let cool in pan for 20 minutes. Remove from pan. Serve warm or at room temperature.