Place butter and 1/3 cup plus 1 tablespoon sugar in a bowl; smear together using a spatula until combined.
Whisk in egg until mixture is creamy.
Add almond flour, salt, vanilla extract, and almond extract. Mix thoroughly.
Cut fruit into wedges if they're larger like plums or apricots. Fill pastry shell with almond paste. Stick fruit snugly, but not too deeply, into filling.
Bake in the preheated oven until browned and almond filling is set, 30 to 40 minutes.
Transfer tart onto a wire rack and let cool completely, at least 40 minutes.
Combine apricot jam and water in a microwave-safe bowl. Bring to a boil in the microwave. Let cool until warm. Brush glaze over cooled tart.