Heat 1/4 water and 3 tablespoons sugar in a saucepan over medium heat until mixture is simmering and sugar is dissolved. Remove from heat and allow syrup to cool to room temperature.
Place raspberries in the bowl of a food processor; pulse until pureed. Pour cooled syrup into raspberry puree and mix well. Strain through a fine-meshed sieve into a container. Cover with plastic wrap and freeze until firm, at least 2 1/2 hours. Break up with a fork and place back in the freezer.
Combine eggs, milk, 1/2 cup water, and 3 tablespoons melted butter in a bowl; beat with an electric mixer until smooth and creamy. Add flour and salt and mix until batter is smooth.
Grease a 9-inch nonstick skillet with 1 teaspoon butter over medium heat. Pour a small ladleful of batter into the skillet. Immediately rotate the skillet until batter coats the cooking surface in a thin layer. Cook until the top of the crepe is no longer wet and the bottom has turned light brown, 1 to 2 minutes. Run a spatula around the edge of the skillet to loosen crepe; flip crepe and cook until the other side has turned light brown, about 1 minute more. Slide ready crepe onto a warmed plate and keep warm. Repeat with remaining batter.
Arrange crepes on plates and dust with confectioners' sugar. Place a scoop of raspberry sorbet on the side and garnish with apple slices, fresh mint leaves, and red currants.