Heat the butter and oil in a medium skillet over medium heat. Add the onions, thyme, 1/2 teaspoon salt and a few grinds of pepper, and cook, stirring occasionally, until the onions are very soft and caramelized, about 15 minutes. Raise the heat a little, and cook to develop a more golden color, about 5 minutes more. Transfer the onions to a small bowl to cool completely.
Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
Cut each slice of ham into 3 strips about 2 inches by 5 inches. Cut the Brie into 24 pieces about 1/4 inch by 1 inch. On each strip of ham, place 1/4 teaspoon of the caramelized onions and 1 piece of Brie, and roll up tightly.
Open the tube of dough, and separate it into 8 triangles at the perforations. Cut each triangle into 3 long, thin triangles (for 24 pieces total). Lay 1 ham roll centered on the tip end of a triangle, roll until you reach the flat end and place, seam-side down, on the prepared baking sheet. Repeat with the remaining triangles and ham rolls. Top each finished roll with a dab (about 1/8 teaspoon) of the remaining onions. Bake until the dough is golden brown and cooked through, 25 to 30 minutes. Serve warm with Dijon mustard on the side.
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