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To prepare the caramelized onions: In a large skillet, melt the butter over medium heat. Add the onions and cook, stirring occasionally, until they soften and begin to turn golden, about 15 minutes.
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Reduce the heat to medium-low and continue cooking, stirring every few minutes, until the onions are deeply caramelized, 40 to 50 minutes.
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Stir in the red wine vinegar, thyme, pepper, and salt. Cook for 1 to 2 minutes more, just until the vinegar is absorbed and the mixture is fragrant.
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Remove from the heat and allow to cool to room temperature before using; caramelized onions can be made up to 5 days in advance. (If making the caramelized onions in advance, store the mixture, covered, in the refrigerator; allow it to come to room temperature before using.)
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To make the tangzhong: In a small saucepan, combine all of the ingredients and whisk until no lumps remain.
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Place the saucepan over low heat and cook the mixture, whisking constantly, until thick and the whisk leaves lines on the bottom of the pan, about 3 to 5 minutes.
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Remove from the heat and transfer to a large mixing bowl, the bowl of a stand mixer, or the bucket of a bread machine (depending on how you'd like to knead the dough).
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To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the cold milk, then the butter and remaining ingredients to the mixing bowl in the order listed; the heat from the tangzhong will help to warm the cold milk and soften the butter.
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Mix — by hand, on low speed of a stand mixer with the dough hook, or in a bread machine set to the dough cycle — to bring the dough together. Next, knead the dough until it’s smooth, elastic, and tacky. This will take up to 15 minutes by hand, 10 to 12 minutes on medium-low speed of a mixer, or the length of the dough cycle in a bread machine.
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Shape the dough into a ball, and let it rest in a lightly greased bowl, covered, for 1 to 1 1/2 hours, until puffy but not necessarily doubled in bulk. While the dough is rising, line an 8" square pan with a parchment sling and lightly grease.
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To add the filling: On a lightly floured surface, gently deflate the dough and roll it into a 23" x 9" rectangle, with the long side facing you.
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Dollop the caramelized onions across the surface and use a small offset spatula to spread them into an even layer. Sprinkle the cheese evenly over the onions, then gently press down with the spatula to adhere.
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Fold the dough in half lengthwise (top edge to bottom), creating a long rectangle that measures 23" x 4 1/2".
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Gently roll, or press with your hands, to widen the dough to 6". Use a sharp knife or bench knife to cut the rectangle crosswise into nine strips, each about 2 1/2" wide.
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Cut each strip almost in half lengthwise, stopping 1/2" before the folded edge, so the two halves remain attached at the top.
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Wrap the two cut pieces around each other to form a loose twist, then roll it up, starting from the cut ends, to form a ball.
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Place each roll into the prepared pan so that the neat side of the twist is facing up; the spot where the two ends of the twist meet should be on the bottom to avoid unfurling during baking. Cover and let the rolls rise for 1 to 1 1/2 hours, until puffy.
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Toward the end of the rise time, preheat the oven to 350°F with a rack positioned in the center.
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To finish and bake the rolls: Brush the rolls with milk, being careful not to deflate the delicate dough.
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Bake the rolls for 28 to 34 minutes, until golden brown on top; a digital thermometer inserted into the center of a roll should read at least 190°F.
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Remove the rolls from the oven and cool them in the pan for 10 minutes before using the parchment sling to transfer them to a rack to cool. Serve warm or at room temperature.
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Storage information: Store leftover French Onion Rolls, well-wrapped, at room temperature for several days; freeze for longer storage.
