Heat oven to 350°F and arrange racks in the lower and upper third of the oven. Combine flour, baking powder, and salt in a large mixing bowl and whisk to break up any lumps.

Combine butter and sugar in the bowl of a stand mixer and beat on medium until light and fluffy, about 4 minutes. Add half the tangerine zest and half the lemon zest and beat until incorporated. Scrape down the bowl then add the eggs in one at a time. Add flour mixture and beat until incorporated.

Scoop into rounded tablespoons and place about 2 inches apart on parchment-lined baking sheets. Bake until edges are lightly golden, but cookies are still soft in the middle, about 12-15 minutes, rotating the baking sheets halfway through. Let cool on the baking sheets for 5 minutes, then remove to wire racks to cool completely.

Whisk together the tangerine and lemon juices, remaining zest, and powdered sugar until smooth. Spread icing over each cookie and let stand at least 1 hour to allow the icing to harden slightly. Store in an airtight container for up to 3 days.