Crush cardamom pods in a mortar and pestle. Remove the green shells and grind the seeds coarsely. Set aside.
Warm the milk to 110-115°F. Add the yeast and a pinch of the brown sugar, whisk, and let sit for 5-10 minutes until foamy.
½ cup + 2 tablespoons whole milk, 2 ¼ teaspoons instant yeast, ¼ cup dark brown sugar
Add the egg, egg yolk, melted and cooled butter, and vanilla to the yeast mixture. Whisk until smooth.
1 large large egg, 1 large egg yolk, 1 tablespoon vanilla extract, 4 tablespoons unsalted butter
In a separate bowl, whisk together the flour, ground cardamom, and salt.
3 cups all-purpose flour, 2 teaspoons cardamom, 1 teaspoon kosher salt
Add about ⅓ of the dry ingredients to the wet mixture and whisk to combine. Add the remaining dry ingredients.
Using a dough hook, knead on low speed for 8-12 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl.
Place dough in a greased bowl, cover, and let rise in a warm spot for 1 hour, or until doubled in size.