Preheat oven to 400°F.
Cook chopped bacon in a large skillet over medium heat until crisp, stirring occasionally; remove and drain on paper towels.
Discard drippings but reserve 2 teaspoons in the skillet.
Add onions, shallots, and sugar to the reserved drippings; cook and stir over medium heat until golden brown, about 15-20 minutes.
Stir in white wine, loosening any browned bits from the pan.
Add thyme, garlic, and pepper; cook uncovered until liquid evaporates, about 2-3 minutes.
Stir in cooked bacon and remove from heat.
On a lightly floured surface, unfold puff pastry and cut into three 9x3-inch rectangles; transfer to a parchment-lined baking sheet.
Brush each pastry with beaten egg white.
Top each with the onion mixture and a slice of goat cheese.
Bake until golden brown, 16-20 minutes.
Cut each rectangle into four appetizers and garnish with additional thyme if desired.