Chocolate-Dipped Shortbread

Arrange racks in upper and lower thirds of oven and preheat to 325°F. Line 2 large cookie sheets with parchment paper.
In electric mixer, beat butter, sugar, vanilla, and salt at medium speed just until smooth. Add flour and mix at low speed until combined. Divide dough in half and shape into 2 disks.
On lightly floured work surface, roll out 1 piece dough to 1/4-inch-thick rectangle. Using fluted pastry wheel or large knife, cut into 2- by 1-inch rectangles. Transfer cookies to baking sheets, spacing 1 1/2 inches apart. Repeat with remaining dough.
Prick each cookie several times with tines of fork, then chill 10 minutes. Bake until edges are golden, about 15 minutes. Cool on pans 5 minutes, then transfer to racks and cool completely.
To decorate: Line baking sheet with clean parchment or wax paper. In medium heatproof bowl set over saucepan of simmering water, melt half chocolate. Add remaining chocolate, remove bowl from heat, and stir occasionally until smooth. Pour into small bowl.
Dip each cookie halfway into chocolate, let excess drip off, and place on baking sheet. Let stand until chocolate is set, about 1 hour. Store in airtight container at room temperature. (Do not refrigerate.)