Line a 9-by-13 inch pan with parchment and spray with cooking spray.
Using some butter, coat the inside of a 4 qt heavy saucepan. Add the sticks of butter, sugar, sweetened condensed milk, corn syrup and salt. Cook over medium-high heat until butter is melted, stirring constantly. Reduce heat to medium.
Insert a candy thermometer into the pan and continue stirring constantly until the temperature is 244ºF (for sea level. check local temperature adjustments according to your elevation), or medium firm-ball stage. Remove from heat.
Stir in vanilla bean paste and pour into prepared pan. Pan will be hot. Transfer to a flat surface and let cool overnight. When caramel sets after a few hours you can wrap the pan in plastic wrap to rest overnight.
Transfer caramel to a clean cutting board and remove any parchment. Cut with a sharp knife into desired sizes. I cut mine about ½ inch deep and 1 ¼ inches long. You will get over 100 pieces.
Sprinkle each piece with flaked sea salt before wrapping well in cellophane or waxed paper. Will keep for up to 2 weeks.
Notes
There's a bit of extra salt (sea salt for a lovely flavor) in the caramel itself, but I also sprinkled gorgeous sea salt crystals on top of each piece before wrapping.