Whisk the flour, baking powder, baking soda and salt in a medium bowl. Beat the butter, granulated sugar and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the egg and vanilla until combined. Reduce the mixer speed to low and beat in the flour mixture until just incorporated.
Transfer half of the dough to a separate bowl and stir in the lemon zest and juice. Beat the molasses, ginger, cinnamon and allspice into the remaining dough. Lay out a large sheet of plastic wrap. Drop alternating spoonfuls of the two doughs onto the plastic to form an 11-by-8-inch rectangle. Use the plastic to roll the dough into a 2-by-11-inch log, twisting the ends closed. Refrigerate until firm, at least 2 hours or overnight.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Slice the dough into 1/2-inch-thick rounds and arrange about 2 inches apart on the prepared baking sheets; sprinkle with sanding sugar. Bake, switching the pans halfway through, until the edges of the cookies are set, 14 to 18 minutes. Let cool 2 minutes on the baking sheets, then transfer to racks to cool completely.