Meanwhile, in a medium bowl, beat the eggs with a whisk. Set beside the stovetop.
In a large saucepan, heat the butter over medium heat, stirring occasionally, until melted and it turns toasty and brown, about 5 minutes. Transfer to a small bowl.
In the same large saucepan, whisk together the brown sugar, golden syrup, heavy cream, and salt. Have an instant-read thermometer ready.
Bring to a boil over high heat. Boil, whisking constantly, until the thermometer registers 235°F (113°C), about 1 minute (this can happen fast). Remove from the heat and whisk in the brown butter, bourbon, and vanilla.
Gradually whisk about half of the warm syrup mixture into the eggs. Then whisk the egg mixture back into the saucepan.