Beat butter and shortening together in a bowl with an electric mixer until smooth; add white sugar, confectioners' sugar, baking soda, cream of tartar, and salt. Beat until just combined. Stir in egg and vanilla extract. Gradually mix in flour until just combined. Cover and refrigerate dough for 3 hours.
Preheat oven to 375 degrees F (190 degrees C).
Roll the dough into 3/4-inch balls and arrange on an ungreased baking sheet. Press your thumb or end of wooden spoon into the center of each dough ball making an indentation. Spoon 1/4 teaspoon jam into each thumbprint.
Bake in the preheated oven until edges of cookies become lightly browned, 8 to 10 minutes. Cool on a wire rack for 5 minutes.
Place white chocolate chips in a microwave-safe bowl; cook in microwave in 10-second intervals, stirring after each interval, until chips are melted. Spoon melted white chocolate into a disposable decorating or plastic bag. Snip off one small corner of bag; drizzle white chocolate over cookies.
