Preheat oven to 375°. Unroll 1 tube of crescent dough into 1 long rectangle; seal seams and perforations. Press onto the bottom of a greased 13x9-in. baking dish. Bake until golden brown, 8-10 minutes.
Layer with half the cheese and all the corned beef. Combine sauerkraut and salad dressing; spread over beef. Top with remaining cheese.
On a lightly floured surface, press or roll second tube of crescent dough into a 13x9-in. rectangle, sealing seams and perforations. Place over cheese. Brush with egg white; sprinkle with caraway seeds.
Bake until casserole is heated through and crust is golden brown, 12-16 minutes. Let stand 5 minutes before cutting.