In a saucepan, cook berries and lemon juice over medium heat until berries release their juices but still hold their shape, about 2 minutes. Stir in sugar and salt; cook until dissolved, 3 to 5 minutes. Strain sauce; reserve whole berries. Let cool.
Line a cupcake tin with plastic wrap, pressing it into cups. Using cookie cutters or an inverted glass, cut out six 2 1/4-inch rounds and twelve 2 3/4-inch rounds from bread.
Pour 1 tablespoon sauce into each cup; add 1 tablespoon berries. Top with smaller bread rounds; add another tablespoon each of sauce and berries. Make another layer of bread, sauce, and berries; top with remaining bread rounds. Press down firmly with the bottom of a glass.
Fold plastic over; cover with a baking sheet. Weight with a heavy skillet. Refrigerate at least 4 hours.
To serve, pull up on plastic wrap, and gently unmold each pudding onto plates; drizzle with sauce.
Using a cupcake tin lined with plastic wrap makes it easy to form, transport, and unmold the chilled puddings.