Position rack in the center of the oven and heat to 400 degrees.
In a medium bowl, mash goat cheese with a fork. Add chives, onion, lemon zest, 1/2 teaspoons kosher salt, and 1/4 teaspoons pepper. Stir until well combined.
On a lightly floured surface, unfold the pastry sheet and lightly dust with flour. Use a rolling pin to roll the sheet into a 12" square. Cut the dough into 9 squares. Put equal of the filling (about 1 tbsp.) onto the center of each square Moisten the edges of a square with a fingertip dipped in water. Fold the dough over to form a triangle, gently pressing to remove air pockets around the filling and pressing the edges of the dough together. Use the tines of a fork to crump and seal the edges of the turnover., Repeat this process with the other dough squares.
Arrange the turnovers on a cookie sheet and bake until the turnovers are puffed and golden all over. Let them cool on a rack for a few minutes and serve warm.
May be filled and shaped up to 2 hours ahead. Cover tightly with plastic wrap or brush with melted butter before refrigerating.