Line an 8-inch square pan with parchment, leaving overhang; lightly grease the paper.
Melt butter in a medium saucepan over medium-low heat. Stir in brown sugar, condensed milk, and corn syrup until smooth.
Bring to a gentle bubble and cook 5–7 minutes, stirring constantly, until thickened and slightly darker. A thermometer should read about 236–240°F.
Remove from heat. Stir in vanilla and fine sea salt, then add white chocolate chips and mix until fully melted and silky.
Scrape the fudge into the prepared pan and smooth the surface.
Dot the top with caramel sauce and swirl with a butter knife. Sprinkle with flaky sea salt.
Chill until firm, about 2–3 hours. Lift out by parchment and cut into small squares.