Beat the butter until smooth and creamy in a large bowl.
Add sugars, golden syrup, and salt. Beat until pale and fluffy.
Mix in the egg until well blended, then add vanilla.
Add flour and mix until combined. Form dough into two disks, wrap in plastic and chill for at least 1 hour.
Preheat oven to 350°F (175°C). Line cookie sheets with parchment paper.
Roll out dough between parchment paper to 1/8-inch thickness for 2-inch cookies or 1/4-inch for 4-inch cookies.
Use a cookie cutter to cut out cookies and transfer them to prepared sheets, spacing 1 inch apart.
If dough becomes too soft, chill in the freezer for 5 minutes.
Bake for 10-12 minutes for small cookies or 12-14 minutes for larger ones, until golden.
Let cookies cool on a sheet for 5 minutes before transferring to racks to cool completely.