1. Crust: Preheat oven to 425 degrees. Whisk together flours, salt, and sesame seeds in a large bowl. Stir in oil and water, then knead until a ball forms. Roll out dough into a 12-inch round and fit it into a 9 1/2-inch tart pan with a removable bottom. Trim any excess length. Prick bottom all over with a fork and bake until crust is golden brown and crisp, about 35 minutes. Let cool on a wire rack.

2. Filling: Reduce oven temperature to 350 degrees. Place spinach in a large pot (if spinach is dry, add 1/4 cup water) and cover. Cook over medium heat, stirring occasionally, until just wilted, about 6 minutes. Transfer to a colander to drain. When cool enough to handle, wrap in a kitchen towel and squeeze to remove excess water. Coarsely chop and transfer to a bowl. Wipe pot dry and heat oil over medium heat. Add shallot, garlic, and red-pepper flakes; cook, stirring, until shallot is softened, about 4 minutes. Transfer to bowl with spinach. Add feta, eggs, and salt; stir until combined.

3. Pour filling into crust and sprinkle with sesame seeds. Bake tart until just set, 30 to 35 minutes. Let cool slightly, about 10 minutes. Serve warm or at room temperature with red-pepper flakes.