In a large skillet over low heat, melt butter with oil. Add onions, rosemary and sugar. Cook, tossing occasionally, until onions are limp and golden brown, 30 to 40 minutes. Stir in the vinegar, scraping any browned bits from bottom of pan.
In a small bowl, whisk together the milk and egg until smooth. Stir in the onions. Preheat oven to 200°C. Line an 11 by 17-inch baking sheet with parchment paper. On a lightly floured surface, roll out pastry to a 9 by 12-inch rectangle. Transfer to baking sheet.
Use a fork to spread onion mixture evenly over pastry. Arrange figs, cut-side up, in even rows on onion mixture. Scatter cheese and pine nuts over figs. Use a pastry brush to dab edges of tart with egg mixture.
Gently fold over edges of tart to form a lip and brush with more egg mixture.
Bake until pastry is puffed and golden, 25 to 30 minutes. Serve, sprinkled with rosemary needles and drizzled with honey, if desired, warm or at room temperature.