In the bowl of an electric mixer fitted with the paddle attachment, cream butter, sugar, and vanilla until combined. On slow speed, beat in flour and salt until combined. Wrap dough in plastic wrap; chill until firm, at least 2 hours, or overnight.
On a lightly floured surface, roll out the dough until 1/16-to-1/8 inch thick. Cut out hearts with a 2-inch heart cookie cutter. Transfer cookies to an ungreased baking sheet; chill for 30 minutes.
Heat oven to 300 degrees with a rack in the center. Press the tines of a fork into each cookie to make designs. Bake until just beginning to brown around the edges, about 18 minutes. (Cookies will keep in an airtight container at room temperature for about 5 days.)
In the top of a double boiler or in a heat-proof bowl set over simmering water, melt the chocolate, stirring occasionally. Stir in the oil. Dip half or one third of each cookie into the chocolate mixture. Transfer cookies to a cooling rack to dry, or place them on a piece of parchment paper in the freezer for 10 minutes. Serve.