In a large bowl stir together 1 ½ cups of the flour and ¼ tsp. salt. Cut in ¼ cup butter until pieces are the size of small peas. In a small bowl stir together water and egg yolk. Add egg yolk mixture to flour mixture. Stir until combined.

Turn dough out onto a lightly floured surface. Knead dough 5 minutes, adding additional flour if dough is sticky. Avoid adding too much flour as it may make the dough stiff and tougher to stretch. Cover with plastic wrap; let rest at room temperature 1 hour.

Cover a large surface (at least 4x3 feet) with a clean cloth or sheet. Lightly flour cloth. On the cloth, roll dough into a 13x9-inch rectangle. Brush with 2 Tbsp. of the melted butter. Cover dough with plastic wrap; let rest 30 minutes.

While strudel dough is resting, prepare apple filling. For filling, in a 10-inch skillet melt butter over medium. Add apples; cook and stir until apples are tender, 5 to 7 minutes. Transfer to a bowl; let stand until cool while stretching the strudel dough. Fold in cheese and sage.

Lightly oil the fronts and backs of your hands. First use your fingers to gently lift and stretch the edges.

To continue stretching the dough, use the backs of your hands and work underneath dough. Starting from the center and working toward the edges, gently lift and pull your hands apart, stretching the dough. Continue pulling dough from the center toward outer edges, working to stretch dough evenly until it is translucent and paper-thin, forming a 32x20-inch rectangle.

Drizzle remaining butter over dough; carefully use a pastry brush to evenly coat.

Preheat oven to 375°F. Line a 15x10-inch baking pan with parchment paper. To wrap and roll strudel, sprinkle nuts evenly over the buttered stretched dough to within 1 inch of the edges.

Beginning 4 inches from one of the 20-inch sides of dough, spoon apple filling across the short side of the dough in a 4-inch wide log to within 1 inch of both long edges.

Using the cloth beneath dough as a guide, gently lift 4-inch section of dough and lay it over filling. Slowly and evenly lift cloth and tightly roll up dough and filling. If needed, cut excess dough from ends to within 1 inch of the filling. Fold ends under to seal.

Carefully transfer the strudel roll to the prepared baking pan. Curve the ends to form a crescent shape. Whisk together egg white and 1 Tbsp. water; brush over top of strudel. (If using phyllo dough, roll up and seal strudel but do not curve roll into a crescent shape. Fold in ends; brush with the 2 Tbsp. melted salted butter instead of egg mixture.)

Bake until golden, 40 to 45 minutes. (If using phyllo, bake 35 to 45 minutes.) Carefully remove strudel from pan. Cool on a wire rack 20 minutes. Using a serrated knife, slice off the ends that don't contain filling; discard. Slice remaining strudel into 2-inch portions. Serve warm.