Preheat oven to 425°F. Mist a pie pan with cooking spray or lightly grease.
Brown butter in a skillet set over medium heat, whisking and keeping an eye on the butter to ensure it doesn’t burn—butter will smell nutty and be the color of hazelnuts when it’s done. Toss in fennel, 1 cup sliced figs, 1/2 teaspoon of kosher salt, and white pepper. Cook until fennel softens, about 3 minutes. Remove from heat and cool.
Meanwhile, lightly beat eggs. Whisk in milk and cream with remaining 3/4 teaspoon salt. Stir in 3/4 of the grated Parmesan, reserving the rest. Add in figs and fennel, stirring to combine. Pour into prepared pie plate. Sprinkle remaining Parmesan on top.
Bake for 10 minutes. Then, turn heat down to 375°F and remove quiche from the oven to scatter fennel fronds and dot remaining fig slices on top. Immediately place it back in the oven to cook for 20 minutes, covering with foil at the 15-minute mark if the top is golden. Quiche is done when set and the middle doesn’t jiggle. Cool for 5 minutes before tipping onto a serving plate, slicing and eating.