Line the bottom and all 4 sides of an 8-inch square baking dish with aluminum foil, then coat lightly with cooking spray. (Alternatively, line with 2 sheets of parchment paper, positioning them perpendicular to each other so that all four sides are covered and the paper extends over the edges by a few inches. Use some cooking spray to hold the parchment in place if needed.)

Place 1 ¼ cups creamy peanut butter, 2 sticks unsalted butter, and ¾ teaspoon kosher salt in a large, wide saucepan. Cook over medium-low heat, stirring frequently, until the butter is melted and the mixture is smooth, 3 to 5 minutes. Remove the saucepan from the heat and stir in ½ teaspoon vanilla extract.

Sift 3 cups powdered sugar into the melted butter mixture. Stir well with a stiff spatula, using the flat part of the spatula (or switch to a whisk if needed) to work out any lumps of sugar, until smooth, combined, and thickened slightly. Don’t worry if there are tiny specks of sugar in the fudge, but it should be as smooth as possible.

Transfer to the baking dish and spread into an even layer with a metal or small offset spatula. Cover the baking dish (do not let the cover touch the surface of the fudge) and refrigerate until firm and sliceable, at least 3 hours or overnight.

Grasping the excess parchment, lift the slab out of the baking dish and place onto a cutting board. Let sit for 5 to 10 minutes so the fudge warms slightly and won’t crumble when sliced, then cut into 36 pieces with a long knife.