Bring sugar and 3/4 cup water to a boil in a saucepan. Do not stir once syrup begins to boil. Heat to 238 degrees F (114 degrees C) or until a small amount of syrup dropped into cold water forms a soft ball that flattens when removed from water and placed on a flat surface. Remove from heat and add the agar-agar. Pour syrup into a glass bowl; let cool briefly.
Beat the syrup using an electric mixer until thick and white. Mix in vanilla extract and salt until the marshmallow mixture is combined.
Mix 2 tablespoons confectioners' sugar and cornstarch together; spread over the bottom of a 9-inch baking pan. Pour the marshmallow mixture on top; let stand until firm, at least 1 hour.
Cut into square marshmallows using a wet knife. Dust with remaining confectioners' sugar.