Prepare the Honey Glaze: Melt butter in a 2- to 3-quart saucepan over medium. Stir in honey, corn syrup, granulated sugar, orange zest, orange juice, and thyme. Increase heat to medium-high, and bring to a boil; cook 1 minute. Remove from heat, and set aside. (Glaze may be refrigerated in an airtight container up to 1 week.)

Prepare the Sausage Filling: Cook sausage in a 12-inch cast-iron skillet over medium, stirring to crumble, until browned and done. Remove from heat; drain well. Beat butter, honey, and brown sugar with a heavy-duty electric stand mixer fitted with the paddle attachment on medium speed until incorporated. Add sausage, and beat until incorporated. Set aside. (Filling may be refrigerated in an airtight container up to 3 days. Bring to room temperature before using.)

Prepare the Bun Dough: Heat milk in a 3-quart saucepan over medium until bubbles begin to form around the edge of pan. Remove from heat. Combine warm water and yeast in a 1-cup measuring cup. Let stand 10 minutes. Add yeast mixture, salt, 2 cups of the flour, and 1 tablespoon of the granulated sugar to warm milk; stir until relatively smooth. Place mixture in a warm place (85°F) until bubbly, 10 to 15 minutes.

Meanwhile, beat shortening with a heavy-duty electric stand mixer fitted with the paddle attachment on medium-high speed until fluffy. Add remaining 2/3 cup granulated sugar; beat until combined. Add eggs, 1 at a time, beating just until blended after each addition. With mixer running on low, gradually add bubbly yeast mix- ture to shortening mixture until fully incorporated. Add remaining 3 cups flour, in 4 batches, beating just until blended after each addition. Replace the paddle attachment with the dough hook; beat dough on medium speed until smooth, 10 to 12 minutes.

Turn dough out onto a floured work surface; knead until very smooth, about 2 minutes. Transfer to a lightly oiled bowl; cover with plastic wrap. Let stand in a warm place (85°F) until dough doubles in size, about 1 hour.

Place 2 1/2 cups of the Honey Glaze in a 12-inch cast-iron skillet. Place dough on a lightly floured surface; punch dough down, and divide in half. Roll 1 dough half out into a 10- x 8-inch rectangle. Spread half of the Sausage Filling over dough rectangle, leaving a 1-inch border. Starting from one long side, roll dough up to enclose filling. Place dough cylinder seam side down; cut off and discard 2 short ends to create smooth ends. Cut cylinder crosswise into 4 to 5 (1 1/2-inch-thick) rounds. Starting in center of skillet and working outward, place rounds, cut side down, on Honey Glaze in skillet. Repeat with second dough half and remaining Sausage Filling. Place skillet in a warm place (85°F); let stand until dough rounds have plumped up and press against each other, 20 to 30 minutes.

Preheat oven to 375°F. Bake honey buns on middle oven rack until golden brown on top, 55 minutes to 1 hour. (If buns are getting too brown, cover with aluminum foil after baking 30 minutes.)

Carefully invert honey buns onto a serving platter. Serve warm or at room temperature, drizzled with remaining Honey Glaze.