Cut four sheets of parchment paper into 15-x-22-inch rectangles. Fold each piece in half crosswise. With a pencil, mark half of a heart on each folded piece, making the fold the center of the heart. Set aside.

Place a large 12-inch skillet with lid over medium heat. Swirl in oil. Add onions and sprinkle with 1/4 teaspoon salt and 1/4 teaspoon sugar. Cover and cook, stirring occasionally, until onions begin to release their liquid and soften, about 5 minutes. Remove cover, reduce heat to medium-low to low and cook, stirring often, till onions are soft and golden, 10 to 15 minutes. Remove from heat. Stir in Figs, chopped sage and wine.

Adjust oven racks to upper and lower thirds of oven; preheat oven to 400°.

Place one chicken breast in a food-safe plastic bag. Seal bag. Place bag on a cutting board. Pound with a meat mallet to an even, 1/3-inch thickness. Repeat with remaining breasts. Cut chicken into 4 equal pieces. Place chicken pieces on a plate. Sprinkle both sides of each with ½ teaspoon salt and black pepper. Place a sage leaf on each and wrap with one slice of prosciutto.

Open parchment hearts. Spoon ¼ of onion-Fig mixture in center of each heart. Spread to an oval shape about the size of a chicken piece. Top with chicken. To seal packets, make small, overlapping folds around the edge of each heart. Press each fold firmly and crease the edge so packets are very tightly sealed and will not open during baking.

Place packets on two large baking sheets (2 packets on each sheet). Place one sheet on upper rack and one on lower rack of oven. Bake for 18 to 20 minutes, switching position of sheets halfway through cooking, until packets are puff and chicken is done (temperature is 165° measured with an instant read thermometer).

Place packets on serving plates. Snip open with scissors. Serve warm.