Using an electric mixer, beat butter in a medium bowl on low speed until smooth, 1-2 minutes.
Add powdered sugar, pepper, and kosher salt. Reduce speed to medium and beat, occasionally scraping down sides of bowl, until light and fluffy, 4-5 minutes.
Add flour and cheese. Reduce mixer speed to low and beat mixture just until dough comes together.
Wrap dough in plastic and flatten into a rectangle. Chill until firm, at least 2 hours.
You can make the dough up to 3 days ahead of baking. Keep chilled. Let stand at room temperature for 30 minutes before continuing.
Place fennel seeds in a resealable plastic freezer bag. Coarsely crush with a rolling pin or the bottom of a skillet. Alternatively, pulse in a spice mill until coarsely crushed. Transfer to a small bowl; stir in sea salt. Set fennel salt aside.
Preheat to oven to 330°F. Line a baking sheet with parchment paper.
Remove plastic wrap from dough. Roll out dough on a lightly floured surface to a 10x8" rectangle about 1/4" thick.
Using a rectangular cookie cutter, cut the dough and arrange cookies on prepared baking sheet, spacing 1" apart.
Brush cookies with oil, then sprinkle with fennel salt.
Bake, rotating sheet halfway through, until cookies are golden brown, 20-24 minutes.
Let cool on baking sheet for at least 10 minutes. Serve warm or at room temperature.
