In the bowl of a stand mixer, whisk together 1 cup (125 grams) flour, sugar, yeast, and salt until combined.

In a small saucepan, combine ⅓ cup (80 grams) water, milk, butter, oil, and honey. Cook over medium heat, stirring frequently, until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add hot milk mixture to flour mixture; using the paddle attachment, beat at medium-low speed until combined. Beat in egg and vanilla until combined. Add 1¼ cups (156 grams) flour; beat at low speed just until combined.

Switch to the dough hook attachment. Beat at low speed until dough just starts to pull away from sides of bowl, 5 to 8 minutes; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if needed. (Dough will still be quite soft and slightly sticky; it will still stick to bottom and slightly up sides of bowl but should not seem excessively wet.)

Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let stand in a warm, draft-free place (75°F/24°C) until doubled in size, about 30 minutes.

Cut out 12 (6×5½-inch) pieces of parchment paper. Place pieces, evenly spaced, on 2 baking sheets; spray parchment with cooking spray.

Punch down dough; cover and let stand for 10 minutes. On a lightly floured surface, roll dough into a 12-inch square (about ¼ inch thick). Sprinkle dough evenly with cinnamon, leaving a ½-inch border along one side. Starting with side opposite border, roll up dough, jelly roll style; pinch seam to seal. Roll seam side down; gently shape to 12 to 12½ inches and even thickness. Using unflavored dental floss, cut dough into 12 slices (about 1 inch thick each). Place 1 slice on each prepared parchment square. Using a piece of plastic wrap placed on top, press rolls to ¼-to flour-inch thickness. Cover dough with sheets of lightly greased plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until puffed and dough holds an indentation when poked, 15 to 20 minutes.

Meanwhile, in a large heavy-bottomed saucepan or Dutch oven, pour oil to a depth of 2 inches, and heat over medium heat until an instant-read thermometer registers 360°F (182°C) to 365°F (185°C). (See Note.)

Line 2 rimmed baking sheets with paper towels. Working in batches of 2, use parchment paper to gently pick up buns and add to oil; using kitchen tongs, remove parchment, and turn buns immediately. Fry until golden brown, 1 to 1½ minutes per side. Using a spider strainer, remove buns from oil, and let drain on prepared pans; let cool slightly. (Gently press unfried rolls again before adding to oil if they get too puffy.)

Transfer buns to wire racks placed over 2 parchment paper-lined rimmed baking sheets. Working one at a time, place 1 bun in Honey Glaze, using a spoon or a small silicone spatula to evenly coat on both sides. Using 2 forks, gently lift bun from glaze, letting excess drip off; transfer to prepared wire rack. Serve immediately, or let stand until glaze is set.