Lightly oil a 9-inch square baking dish with cooking spray. Dust generously with 1/2 cup confectioners' sugar.
Bring 1/2 cup water to a boil in a small saucepan. Remove from heat and stir in agar agar; let soak until dissolved, 10 to 15 minutes.
Beat egg whites in a stand mixer fitted with the whisk attachment until soft peaks form.
Combine remaining 3/4 cup water, white sugar, and corn syrup in a small saucepan. Cook over medium-high heat until mixture registers 250 to 265 degrees F (121 to 129 degrees C) on a candy thermometer or until a small amount of syrup dropped into cold water forms a rigid ball. Remove from heat; stir agar agar mixture into the hot syrup.
Turn stand mixer on to continue beating egg whites. Pour hot syrup in a thin stream down the side of the bowl, beating until egg whites are very stiff. Stir in vanilla extract.
Spread marshmallow mixture evenly in the prepared pan. Dust with remaining 1/2 cup confectioner's sugar. Let stand until set, 8 hours to overnight.
Dip a sharp knife in very hot water and wipe off; slice marshmallow mixture into 2-inch pieces.