Cut fronds from fennel bulbs; reserve. Cut stalks from bulbs and finely chop. In a bowl, stir together chopped stalks, coarse sea salt, and sugar. Place salmon in a 7-x-7-x-2-inch or 8-x-8-x-2-inch glass baking dish; cover on all sides with fennel mixture, pressing to adhere. Cover with plastic wrap, pressing wrap directly onto top and sides of fish. Place heavy cans on top of fish. Refrigerate 2 to 3 days.

When ready to serve, use a ring-shaped cutter to cut a circle from center of each bread slice. Using a rolling pin, slightly flatten bread circles and outer rings. In a large nonstick skillet, heat 1½ tablespoons butter over medium heat until melted. Add inner circles of bread and cook, turning, until golden on both sides, about 5 minutes; transfer circles to a plate. Repeat with 1½ tablespoons butter and outer rings of bread, toasting in batches, if necessary. Remove skillet from heat and reserve.

Scrape curing mixture from salmon (some will remain). Using a knife with a long thin blade, thinly slice salmon diagonally at a 45-degree angle from top of fillet toward skin.

Trim fennel bulbs and thinly slice. Add remaining tablespoon butter and oil to skillet and heat over medium heat until butter is melted. Add fennel slices and generous pinch salt; cook, stirring occasionally, until wilted, 7 to 10 minutes. Remove from heat.

Divide bread circles among 4 plates. Cover with alternating fennel and salmon slices. Top with bread rings. Stir together mayonnaise and dill. Chop reserved fennel fronds; sprinkle fronds and dill over sandwiches and serve with prepared mayonnaise.