Line an 8-inch square pan with parchment paper, leaving overhang on two sides; set aside.
In a heavy 3-quart saucepan, combine light brown sugar, evaporated milk, unsalted butter, and fine salt. Cook over medium heat, stirring gently until the sugar dissolves and the butter melts, 3–5 minutes.
Clip on a candy thermometer. Increase heat to medium-high and bring to a steady boil. Do not stir while boiling. If crystals form on the sides, brush them down with a wet pastry brush. Cook until 236–240°F (soft-ball stage), 8–12 minutes; a drop in cold water should form a soft, malleable ball.
Remove from heat and let the mixture cool, undisturbed, to 110–120°F, 30–45 minutes; the pan should feel just warm and the surface slightly thickened.
Add pure vanilla extract. Beat vigorously with a wooden spoon or a hand mixer on low until the fudge thickens, lightens in color, and loses most of its gloss, 3–7 minutes.
Fold in the chopped walnuts just until evenly distributed.
Immediately scrape the fudge into the prepared pan and smooth the top; avoid scraping any sugary residue from the sides of the pot into the pan.
Let stand at cool room temperature until firm, 1–2 hours. Lift out using the parchment, cut into 36 squares, and store airtight at cool room temperature for up to 1 week.