In the bowl of an electric mixer fitted with a paddle attachment, mix together the butter and 1 cup of sugar until they are just combined. Add the vanilla. In a medium bowl, sift together the flour and salt, then add them to the butter-and-sugar mixture. Mix on low speed until the dough starts to come together. Dump onto a surface dusted with flour and shape into a flat disk, about 1-inch thick. Wrap in plastic and chill for 30 minutes, until firm but still pliable.
Roll the dough 1/2-inch thick on a lightly floured surface and cut with a 3-by-1-inch finger-shaped cutter. Place the cookies on an ungreased baking sheet and sprinkle with sugar. Bake for 20 to 25 minutes, until the edges begin to brown. Allow to cool to room temperature.
When the cookies are cool, place them on a baking sheet lined with parchment paper. Put 3 ounces of the chocolate in a glass bowl and microwave on high power for 30 seconds. (Don't trust your microwave timer; time it with your watch.) Stir with a wooden spoon. Continue to heat and stir in 30-second increments until the chocolate is just melted. Add the remaining chocolate and allow it to sit at room temperature, stirring often, until it's completely smooth. If there are still bits of unmelted chocolate after about 5 minutes, return to the microwave in 5-second intervals, stirring, until completely smooth. Stir vigorously until the chocolate is smooth and slightly cooled; stirring makes it glossier.
Drizzle or dip 1/2 of each cookie with just enough chocolate to coat it. Place the cookie back on the parchment-lined baking sheet and let stand at room temperature until the chocolate has firmed up, about 20 minutes.
Cook's Note: If a 3-by-1-inch cookie cutter is not available, you can cut the shortbread into rectangles using a fluted pastry wheel for decorative edges or simple cut with a butter knife.