In small bowl, soak currants in warm water for 30 minutes. Drain and set aside.
Rinse rice under cold running water; drain and set aside.
Scrub mussels under cold running water; trim off any beards. Discard any with cracked shells or that do not close when tapped.
In Dutch oven, bring 1/2 cup water and 1 tbsp of the oil to simmer over medium-high heat; cook mussels until they open and mussels just begin to pull away from shells, about 4 minutes. Discard any mussels that have not opened. Using tongs, transfer mussels to colander set over bowl. Pour mussel cooking liquid into measuring cup and add enough water to make 1 cup.
Heat remaining oil in saucepan over medium heat; cook pine nuts, stirring, until golden brown, about 1 minute. Add rice, tomato and onion; cook for 2 minutes, stirring constantly. Stir in stock, reserved mussel liquid, currants, cinnamon, salt, pepper and sugar; bring to boil. Cover and simmer over low heat until rice is tender and liquid is absorbed, about 20 minutes. Remove from heat; add dill and parsley. Fluff rice, incorporating herbs. Let cool.
Being careful not to break shells apart, use spoon to gently remove mussels. Arrange open shells on work surface. Place 1 mussel in each bottom shell; stuff shell with scant tablespoonful of the rice mixture. Squeeze shells together and place in Dutch oven, mussel side down. Cover with circle of parchment paper; weigh down with plate topped with 4 large cans. Avoiding stuffed mussels, pour reserved mussel liquid down side of pan. Simmer over medium heat until cooking liquid has evaporated, about 20 minutes.
Transfer mussels to serving platter; let cool to room temperature. Serve as is, or open mussels wide and gently loosen oval of stuffing with knife, inverting stuffing onto other shell so mussel shows on top. Garnish with lemon wedges.