How to Make It
Preheat oven to 350°F. Bring water to a boil in a large Dutch oven. Add 1/4 cup of the salt, stirring until dissolved. Add chard; cook 2 minutes. Drain chard; spread in an even layer on a baking sheet lined with paper towels. Let cool 5 minutes. Squeeze remaining water out of chard; pat dry with paper towels.
Heat oil and 1 tablespoon of the butter in a large nonstick skillet over medium-high. Add onions; cook, stirring often, until golden, 7 to 9 minutes. Add scallions and garlic; cook, stirring constantly, 1 minute. Add chard; cook, stirring often, until wilted and tender, about 7 minutes. Add parsley, mint, and oregano; cook, stirring constantly, 1 minute. Remove from heat, and transfer mixture to a large bowl. Cool slightly, about 10 minutes. Stir in feta cheese, Parmigiano-Reggiano cheese, pine nuts, pepper, and remaining 1 1/4 teaspoons salt. Set aside.
Melt remaining 1/2 cup butter in a small saucepan. Fit 1 phyllo sheet into a lightly greased 9 1/2-inch deep-dish pie plate, allowing edges to evenly hang over sides. Brush with a small amount of melted butter. Place another phyllo sheet directly on top of the first, and brush with butter. Rotate pie plate slightly, about 45°, and repeat process with an additional 2 phyllo sheets and butter. Repeat procedure with remaining phyllo sheets, rotating pie plate about 45° after every 2 phyllo sheets.
Spread chard mixture evenly over phyllo. Brush phyllo edges around the pie plate with melted butter; fold edges up over chard filling toward the center, gently pleating as you go. (Phyllo will only partially cover filling.) Brush top of phyllo with remaining melted butter. Bake in preheated oven until crust and filling are golden brown, about 40 minutes. Remove from oven; cool on a wire rack 10 minutes, and serve.