1. Heat oven to 400 degrees F. Melt butter and oil in 12-inch straight- sided skillet. Add onions and salt; cook, covered, 10 minutes, stirring. Uncover; cook over low heat 30 minutes to caramelize.
2. Meanwhile, on floured surface, unfold pastry. Roll lengthwise to form 12 x 9-inch rectangle. Cut three 1-inch-wide strips from long edge, leaving 12 x 6-inch rectangle. Transfer rectangle to ungreased baking sheet. Brush edges with beaten egg mixture. Cut one strip in half for two 6 x 1-inch strips; place strips on short sides of rectangle. Place remaining 2 strips on long sides. Pierce bottom of pastry at 1/2-inch intervals with fork. Bake at 400 degrees F for 15 to 20 minutes, until golden.
3. After onions are golden or caramelized, stir in wine, scraping up any browned bits from bottom. Cook until wine is evaporated, 2 minutes. Add rosemary. Spread onions into tart; top with cheese. Bake 3 to 4 minutes, until cheese is melted.