Russian fur traders from across the strait introduced salmon to Kodiak Island in the 18th century, and it has since remained a staple fish in Alaskan homes. This dish is called Kulebyaka in Russia, but in the northern region of Alaska it's pirok, perok or peroche--all variations of the word 'priog', the Russian word for pie. It's pure "comfort food" and is a favorite dish at Alaska's Winterlake Lodge near the Iditarod Trail and at Tutka Bay Lodge near Homer. It's also a popular dish in the coastal regions of Russia where commercial salmon fishing is predominate.
Melt butter in a large nonstick skillet over medium-low heat. Add onion and cook, stirring occasionally, until tender, about 7 to 8 minutes. Stir in mushrooms, cabbage and vinegar; increase heat to medium. Cover pan and cook 4 minutes; uncover, toss and cook 2 more minutes. Remove vegetables from pan, season with salt and pepper to taste, and set aside.
Wipe out skillet, add olive oil and set over medium-high heat. Add salmon and season lightly with salt and pepper. Cook salmon 5 minutes per side; remove to a plate and let cool. Flake salmon into large chunks and set aside.
Set a sheet of puff pastry on a lightly floured surface. Gently roll out until it is large enough to fit a 9-inch deep-dish pie plate. Transfer pastry to pie plate, allowing extra dough to drape over edge.
Spread rice over bottom of pastry. Evenly distribute hard-boiled egg over the rice, followed by flaked salmon. Sprinkle with cheese, then bread crumbs. Top with the vegetable mixture, sprinkle with parsley and drizzle cream over top.
Roll out remaining sheet of puff pastry on a lightly floured surface until it is large enough to cover pie. Brush rim of bottom pastry with water and place second sheet of pastry directly on top. Using kitchen scissors or a paring knife, trim off excess dough. Use a fork to crimp the edges together to help seal the pastry.
Cut a few small slits in the top of the pie to allow steam to escape. Brush top of pie with beaten egg. Bake until pastry is puffed and golden brown, about 35 to 40 minutes.