Slice the bread into thick slices, about 1½ to 2 inches.
In a large, shallow bowl, whisk together the eggs, milk, brown sugar, cinnamon, kosher salt, and vanilla extract until completely smooth.
Dip each slice of bread into the egg mixture, letting it soak for a few seconds per side. Let the excess drip off.
Grease the air fryer basket with cooking spray or line it with perforated parchment paper. Arrange the dipped slices in a single layer without overlapping, working in batches as needed. Brush the tops lightly with melted butter.
Air fry at 330 degrees F for 8–12 minutes, flipping halfway through and brushing the second side with butter, until golden brown and the custard is set in the center.
While the French toast cooks, combine the brown sugar, heavy whipping cream, and corn syrup in a small saucepan. Bring to a gentle simmer over medium heat, whisking until smooth, 3–5 minutes. Remove from the heat and let cool slightly; the syrup thickens as it sits.
Serve the French toast warm, drizzled with caramel syrup and topped with fresh strawberries, whipped cream, and a dusting of powdered sugar.