In a stand mixer or large bowl, mix flour and yeast. Heat milk, sugar, salt, and 2 tablespoons butter until very warm (120° to 130°F). Add to flour mixture with egg. Use a dough hook or electric mixer to beat on medium speed until smooth and elastic, 4 to 5 minutes. Scrape out to a lightly floured surface and shape into a ball. Place into an oiled bowl, cover, and set aside until doubled, about 1 hour.

Melt 3 tablespoons butter in a large skillet over medium heat. Add onions and cook until soft, about 5 minutes. Reduce to very low and cook until onions are golden, 35 to 45 minutes. Remove from the heat and add salt to taste.

Grease a 10-to-12-inch Bundt pan. Mix cheese, cherries, and thyme. Punch dough down and stretch or roll out to a 9-inch square. Melt remaining 3 tablespoons butter and brush dough with 1 tablespoon. Cut dough into 1 1/2-inch squares. Arrange 1/3 dough in bottom of prepared pan buttered side down. Top with 1/3 cheese-cherry mixture and 1/3 onions. Repeat 2 more times. Drizzle with remaining 2 tablespoons melted butter, cover loosely, and set aside 20 minutes.

Preheat oven to 400°F. Bake loosely covered with foil for 25 minutes. Remove foil and bake until golden, 7 to 10 minutes. Cool in the pan for 10 minutes and serve warm.