Line an 8-inch square baking pan with parchment paper and lightly grease the parchment.
In a heavy-bottomed saucepan, combine the butter, brown sugar, granulated sugar, heavy cream, sweetened condensed milk, corn syrup, and fine sea salt.
Cook over medium heat, stirring constantly, until the butter melts and the mixture is completely smooth.
Attach a candy thermometer and continue cooking, stirring frequently, until the mixture reaches 236°F to 238°F.
Remove the pan from the heat and let the mixture cool undisturbed until it reaches 120°F to 125°F.
Stir in the vanilla extract.
Beat with a wooden spoon or hand mixer on low speed until the mixture thickens, lightens slightly in color, and loses most of its glossy appearance, about 2 to 4 minutes.
Immediately spread the fudge into the prepared pan and smooth the top.
Sprinkle with flaky sea salt.
Let the fudge cool completely at room temperature.
Refrigerate for 1 hour to fully set.
Lift from the pan and cut into 36 squares.