Position an oven rack in the center and preheat to 425°F. Place a 10-inch cast iron skillet inside while the oven heats — the screaming hot pan is what creates that dramatic puff.
Make the pistachio crumble: Toss chopped pistachios, oats, brown sugar, melted butter, a pinch of cardamom, and a pinch of flaky sea salt together on a small parchment-lined baking sheet. Set aside — this will go into the oven later.
Prep the stone fruit: Toss peach wedges with honey, melted butter, vanilla, and a pinch of sea salt. Arrange cut-side up on a separate parchment-lined baking sheet. Set aside.
Make the batter: Blend the eggs in a blender on high for 30 seconds until frothy and pale. Add milk, flour, sugar, cardamom, cinnamon, salt, and vanilla. Blend for another 30 seconds until completely smooth with no lumps. The batter should be thin — thinner than traditional pancake batter. Let it rest while you brown the butter.
Brown the butter: Carefully pull the hot skillet from the oven using thick oven mitts. Place on the stovetop over medium heat. Add the 4 tbsp butter and swirl constantly as it melts, foams, and turns golden amber with a nutty aroma — about 2-3 minutes. Watch closely; it goes from browned to burned fast. You should see dark golden flecks on the bottom.
Work quickly: Swirl the brown butter to coat the entire bottom and sides of the skillet. Immediately pour the batter into the center of the hot skillet — do not stir. Slide the pan into the 425°F oven immediately.
Bake the Dutch baby for 18-22 minutes until dramatically puffed, deeply golden on the edges, and set in the center. Do not open the oven door for the first 15 minutes — the steam is what creates the puff.
While the Dutch baby bakes, slide the stone fruit into the oven on a separate rack during the last 12-15 minutes of baking. Add the pistachio crumble tray during the last 6-8 minutes until the nuts are toasted and fragrant.
Remove the Dutch baby from the oven — it will deflate slightly, which is completely normal and creates a natural bowl shape. Immediately arrange the roasted peach wedges in the center, scatter the pistachio crumble over the top, dust with powdered sugar, and add a dollop of crème fraîche. Serve straight from the skillet while it's still puffed and steaming.