Melt butter in a medium saucepan over medium-low heat, skimming foam that rises to the surface. When foam subsides, remove pan from heat and carefully pour clarified butter into a large mixing bowl, leaving milky solids behind. Refrigerate clarified butter until almost solid, about 40 minutes.
Add shortening to the clarified butter and beat with an electric mixer on medium-high speed until smooth and pale yellow, about 1 minute. Gradually add sugar while beating constantly, until mixture is light and fluffy, about 1 minute. Reduce speed to low and gradually add flour while beating constantly, stopping mixer to scrape down sides of bowl with a rubber spatula as needed, until flour is fully incorporated and mixture is crumbly, about 1 minute.
Turn dough out onto a lightly floured surface and knead until silky smooth (dough will be soft and pliable but not sticky). Cut dough into 4 equal pieces, loosely cover with plastic wrap, and set aside at room temperature for about 30 minutes.
Preheat oven to 350º. Use your hands to roll dough pieces out on a clean surface into 1 1⁄2"-thick ropes. Cut ropes on the diagonal at 1" intervals, then gently press 1 pine nut on top of each diamond-shaped piece, flattening dough slightly, and transfer to ungreased cookie sheets about 1⁄2" apart. Bake cookies until very pale golden on the bottom, 10-12 minutes. Remove cookies from oven and set aside to cool completely on the cookies sheets. Store in an airtight container for up to 1 week.