In a small bowl, stir together olive oil, dill, salt, sugar, and pepper. Trim fillets to the same size and thickness. Line an 8- by 8-in. baking dish with plastic wrap and lay one fillet, skin side down, on bottom. Cover with dill mixture and top with second fillet, skin side up. Cover fillets with plastic wrap (do not seal edges) and a plate just big enough to fit inside the baking dish. Top plate with a 2- to 4-lb. weight (such as a 6-pack of soda). Cure fish, chilled, 2 days, turning over and draining excess liquid after 1 day.
Remove fillets, scrape off dill, rinse well, and pat dry. Thinly slice halibut on a steep angle, leaving skin behind and creating wide slices.
Whisk together mustards and brown sugar and set aside.
Stack bread slices and cut into 4 equal squares. Top each square with a dollop of crème fraîche. Arrange 1 or 2 pieces halibut over the crème fraîche and top with several dots of mustard sauce and a sprig of dill.
Note: Nutritional analysis is per serving.