Line an 8-inch square pan with parchment, leaving overhang on two sides for easy lifting. Lightly butter or spray the paper and pan sides.
In a small bowl, stir the peanut butter with the vanilla. Microwave on 50% power for 20–30 seconds to loosen slightly; set aside.
Add sugar, evaporated milk, corn syrup, and butter to a heavy 3-quart saucepan. Warm over medium-low heat, stirring gently, until the sugar dissolves and the mixture looks uniform, about 3 minutes.
Clip on a candy thermometer and raise heat to medium. Cook without stirring, swirling the pan once or twice, until the syrup reaches 234–238°F (soft-ball stage), about 6–8 minutes.
Remove from heat and let the bubbling subside for 30–60 seconds. Add the warmed peanut butter mixture and beat briskly with a sturdy spoon for 45–90 seconds, just until the fudge turns glossy to matte and thickens slightly.
Scrape into the prepared pan, spreading evenly. Tap the pan on the counter to release air bubbles, then sprinkle the top with the flaky sea salt. Cool at room temperature until set, about 3 hours (or cool 30 minutes, then refrigerate 30–45 minutes to speed it up).
Lift out using the parchment overhang and cut into 36 small squares, wiping the knife with a warm, damp towel between cuts for clean edges.