Heat a large nonstick skillet over medium heat. Coat pan with cooking spray. Add onion; cook 12 minutes or until browned, stirring frequently. Add chard; cook 5 minutes or until chard wilts, stirring frequently. Remove from heat; stir in 1/4 teaspoon salt. Cool.
Combine remaining 1/4 teaspoon salt, milk, and next 5 ingredients (through bread) in a large bowl; toss well to coat. Place half of bread mixture in an 11 x 7-inch baking dish coated with cooking spray. Top evenly with tofu, half of chard mixture, and 1/2 cup cheese. Top evenly with remaining bread mixture, remaining chard mixture, and remaining 1/2 cup cheese. Cover with foil coated with cooking spray. Refrigerate 8 hours or overnight.
Preheat oven to 350°.
Remove strata from refrigerator; let stand at room temperature 15 minutes. Bake strata, covered, at 350° for 30 minutes. Uncover and bake an additional 15 minutes or until set. Let stand 10 minutes before serving.