Preheat the oven to 400 degrees F. Combine the cream cheese, smoked salmon, peperoncini, shallot, lemon zest, and dill in a small bowl. Blend well with a fork and season with salt and pepper, to taste. You should have about 2/3 cup.
Cut each salmon fillet horizontally down the center through its thickness, almost to the other edge but not quite through. Open each fillet like a book and flatten. Season with salt and pepper. Place a quarter of the cream cheese mixture in the center of one half and cover with the other side to enclose the filling. Season the outside with salt and pepper. Refrigerate while preparing the phyllo.
Arrange 1 sheet of the phyllo on a large work surface. Brush with butter and top with another sheet of the dough. Repeat the procedure with the remaining phyllo dough and butter. Cut the stacked slices of phyllo into 4 pieces. Arrange a piece of the filled salmon in the center of each piece. Drizzle the salmon with equal amounts of the olive oil and lemon juice. Fold in the sides of the phyllo to enclose the fillet. Place the packages seam side down on a baking sheet. (The recipe can be prepared up to 4 hours ahead at this point. Keep refrigerated, covered with plastic wrap, until ready to bake.)
Lightly brush the top of the packages with olive oil. Bake until lightly browned, 18 to 20 minutes.