In a medium bowl, whisk together 1/4 cup plus 2 1/2 tablespoons all-purpose flour, 3/8 teaspoon baking powder, and 1/8 teaspoon kosher salt. Set aside.
In the bowl of a freestanding electric mixer fitted with a whisk attachment, combine 1 jumbo egg and 3 tablespoons plus 2 teaspoons granulated sugar. Mix on medium-high speed for about 1 minute, before increasing the speed to high and whipping for about 4 minutes, until the batter lightens and doubles in volume.
While the egg and granulated sugar is whipping, heat 4 tablespoons unsalted butter, 2 teaspoons dark brown sugar, and 1 1/8 teaspoons clover honey in a small saucepan over medium-high heat, whisking constantly to dissolve the brown sugar. This should take around a minute; once the sugar has dissolved, remove from heat.
Remove the bowl from the mixer stand (from the 2nd step) and use a rubber spatula to fold in half the dry ingredients (from the 1st step); then, fold in the remaining dry ingredients until just combined. Scrape the bottom of the bowl to incorporate any dry ingredients that may have settled there. Pour the warm butter mixture (from the 3rd step) over the batter ad fold until the mixture is incorporated and the batter is smooth.
Place 3 tablespoons pistachio paste in a medium bowl — if the pistachio paste is very stiff, warm in the microwave for about 20 seconds to soften. Use a rubber spatula to stir one-quarter of the batter into the pistachio paste to lighten it, then fold into the remaining madeleine batter. Place the batter in a covered container and refrigerate overnight.
Once the batter has chilled overnight and you're ready to serve the cakes, preheat the oven to 350 (F). Prepare your madeleine pan by spraying generously with cooking spray.
Spoon the refrigerated batter into the madeleine pan using a 1-tablespoon cookie scoop — I filled each cavity with 1 generous tablespoon of batter. Tap the bottom of your madeleine pan against your work surface to smooth the top of the batter. Bake for 8 to 9 minutes in the preheated oven, or until the tops are lightly browned and a skewer inserted in the center of a madeleine comes out clean. The bottoms of the madeleines will brown more quickly than the tops, so keep the tops on the lighter side. Immediately unmold the madeleines and cool on a cooling rack.