Preheat oven to 325°F. Using an electric mixer, beat butter, sugars and salt for about 1 minute. Beat in vanilla. Reduce speed and add flour in 3 additions, beating just until blended. Stir in 3/4 cup chocolate chips.
Transfer dough to an ungreased 13-by-9-inch baking pan. Press dough evenly on bottom of pan with your fingers. Bake until golden brown, about 35 minutes; do not underbake. Let cool for 5 minutes.
Cut shortbread in pan, lengthwise and crosswise into quarters, then lengthwise into 32 rectangles. Let cool in pan.
Melt remaining chocolate and oil in a small saucepan over low heat, stirring, until smooth. Dip half of each cookie into chocolate; place on wax paper. Let stand until chocolate is set. (Cookies may be made up to 4 days ahead and stored, tightly wrapped, at room temperature.)