1. In a food processor or large bowl, combine flour, granulated sugar, and salt. Cut 1/2 cup butter into pieces and add to flour mixture; pulse motor, cut in with a pastry blender, or rub in with your fingers until mixture resembles coarse meal. With motor running (or stirring with a fork after each addition), add egg yolk and 3 to 4 tablespoons cold water, 1 tablespoon at a time; process or stir just until mixture comes together in a ball. Form dough into a flat disk, wrap in plastic wrap, and chill until firm but still pliable, about 1 hour.
2. Meanwhile, spread walnuts in a baking pan and bake in a 375° oven until barely golden under skins, 6 to 8 minutes (leave oven on). Coarsely chop nuts.
3. Peel and core apples; cut each into eight wedges. In a 10- to 12-inch nonstick frying pan over medium heat, melt remaining 2 tablespoons butter. When it's foamy, add apples and stir often until slightly softened and brown at edges, 10 to 12 minutes. Sprinkle brown sugar and nutmeg over fruit and stir until liquid is syrupy and bubbling, about 5 minutes. Stir in walnuts. Remove from heat.
4. Unwrap dough. On a lightly floured surface, with a lightly floured rolling pin, roll into a round about 15 inches in diameter. Line a 12- by 15-inch baking sheet with cooking parchment (or butter the sheet well) and carefully transfer dough round to sheet (edges will hang over sheet).
5. Pour apple mixture onto center of pastry, mounding wedges in a circle about 8 inches wide and 2 inches high. Gently fold edges of dough over apples, pleating as you go, leaving an opening about 4 inches wide in the center. Brush pastry all over with beaten egg.
6. Bake in 375° oven until pastry is golden brown and apples are tender when pierced, 40 to 45 minutes (35 to 40 in a convection oven). Transfer galette (with parchment, if using) to a wire rack to cool. Transfer to a large plate, gently pulling parchment from under tart. Serve slightly warm or at room temperature, cut into wedges.