Preheat oven to 400 degrees F (205 degrees C). Place tart shells on a baking sheet.
Bake in the preheated oven until edge of crust is golden, about 10 minutes.
Lower oven temperature to 350 degrees F (175 degrees C). Position a rack in the upper 1/3 of the oven.
Whisk brown sugar, golden syrup, butter, eggs, vanilla extract, and salt together in a bowl until smooth. Fill pastry shells with sugar mixture about 2/3-full.
Bake in the preheated oven until filling foams and is bubbly and golden, about 15 minutes. Remove pan and cool for a few minutes on a wire rack; serve warm or at room temperature.