In a small bowl, soak the currants in the wine for at least 4 hours but preferably overnight.
Heat the oil in a 10-inch straight-sided saute pan over medium-high heat. Add the rice, onion, pine nuts, scallions, lemon zest, garlic, and 3/4 tsp. salt. Reduce the heat to medium, and cook, stirring, until the onion is softened and fragrant, 3 to 4 minutes. Add the currants with their liquid and 1/2 cup of water. Cook over medium-high heat until most of the liquid has evaporated, about 6 minutes. Transfer the mixture to a bowl and stir in the chopped dill and mint.
Wipe the pan clean and line the bottom and sides with about 15 of the extra grape leaves—choose leaves that are large, torn, or have many holes. Top with half of the reserved dill and mint stems.
On a large cutting board or clean counter, lay out as many grape leaves as you have room for, veined side up. Remove any long stems. Drop a scant tablespoon of the rice mixture onto the bottom center of each leaf and with your fingers, shape the filling into a mound about 2 inches wide and 1 inch tall. Fold the bottom of the leaf up over the filling and then fold in the sides of the leaf to enclose the filling. Using the side of your finger, gently press the filling down toward the bottom fold of the grape leaf, and then tightly roll the filling up in the rest of the leaf, forming a stubby, compact cylinder.
As you fill and roll the leaves, tightly pack them in the pan, seam side down, forming concentric circles. Keep the layer of stuffed leaves as even and flat as you can.
Continue stuffing and packing the leaves until all the stuffing is used. Scatter the remaining dill and mint stems over the stuffed leaves and then cover with the remaining extra grape leaves. Fold over the grape leaves from the sides to completely cover the stuffed grape leaves.
Fill the pan with enough water to barely cover the leaves. Sprinkle the lemon juice on top. Weight down the stuffed leaves with a heavy heatproof plate or a pan partially filled with water. If you're using a plate that's not very heavy, put a 15-oz. can on the plate. Bring to a boil over medium-high heat and then reduce the heat to medium low and simmer for 50 minutes. Carefully remove and taste one stuffed leaf to see if the rice is soft. If not, continue cooking, checking every 10 minutes, until soft. Remove the pan from the heat and let cool at room temperature until warm, about 1 hour.
Remove the plate or pan from the top of the dolmades and carefully pour off the liquid, gently pressing down on the dolmades with a slotted spatula. Cover and chill for at least 1 hour. Serve cool or at room temperature.
You can make the stuffed grape leaves a day ahead.