Remove small bones from salmon with tweezers. Rinse salmon with aquavit.
Combine sugar and next 3 ingredients. Pour half of sugar mixture into a 13- by 9-inch baking dish. Place salmon fillet over sugar mixture, skin side down; top with remaining sugar mixture and dillweed sprigs. Cover with plastic wrap. Place another 13- by 9-inch baking dish over plastic wrap. Fill with heavy cans. Chill 24 to 48 hours. Turn fillet every 12 hours, draining off any liquid that collects and replacing weights.
Rinse coating from salmon; pat dry. Slice thinly, starting at the tail. Serve on toasted French bread with Honey-Dijon Sauce. Garnish, if desired.