Preheat oven to 350 degrees. In the bowl of a stand mixer on low speed, using paddle attachment, combine flour, sugar, baking powder, and salt. Add butter and beat on medium until coarse crumbs form. With mixer on low, add eggs, one at a time, beating after each addition; mix in apricots, pistachios, and fennel seeds. Continue to mix until dough forms a ball, about 1 minute.
On a parchment-lined baking sheet, shape dough into two 2-by-12-inch logs. Bake until dry and set, about 25 minutes. Let cool completely on sheet on a wire rack. Transfer to a cutting board. With a serrated knife, cut logs on the diagonal into 1/4-inch slices. Bake on parchment-lined sheets until golden, 15 to 20 minutes, rotating sheets halfway through. Let cool completely on sheets on racks. (Store in airtight containers, up to 1 week.)