Make the brioche dough: In the bowl of a stand mixer or in a large mixing bowl, mix together the yeast with the milk. Add the egg, butter, flour, salt, sugar, and cinnamon and mix until the ingredients are just barely combined. Let the mixture stand for 5 minutes so the flour has time to hydrate.
Using the hook attachment, beat on medium speed until a soft dough forms. Increase speed to medium-high and beat until the dough is soft and supple, about 6-8 minutes longer. *If you do not have a stand-mixer with a hook attachment, knead the dough by hand in this step.*
Transfer the dough to a lightly floured surface and knead it with your hands for 1 minute. Form the dough into a ball and transfer it to a lightly greased bowl. Cover the dough loosely with plastic wrap, a paper towel, or aluminum foil and let sit in a warm place until doubled in size, about 45 minutes.
Make the sautéed apple filling: In a large nonstick skillet, add the apples, butter, cinnamon, nutmeg, and cardamom and stir until evenly coated. Cook over medium-high heat until the apples begin releasing their juices and are tender. Set aside to cool.
Make the cinnamon sugar filling: Add the softened butter, dark brown sugar, and cinnamon to a medium bowl and combine using a fork until a thick paste forms. Set aside.
Line a 9 x 5-inch pan with parchment paper, leaving an overhang on the two long sides. Turn the risen dough out onto a lightly floured work surface, punch it down and, using a rolling pin, roll into a 12×16-inch rectangle. Make sure the dough is smooth and evenly thick.
Spread the cinnamon sugar filling all over the dough, then spread the sautéed apples on top of the filling, gently patting them down into the filling. Starting with the shorter end, tightly roll up the dough to form a 12-inch-long log. Using a sharp knife, cut the log in half, lengthwise. Turn the dough slightly to ensure the cut sides are facing upwards on both pieces. Pinch the ends together on one side then gently twist the two pieces of dough, pinching the other ends together as well. Carefully push on both ends to compress the twisted strands of dough until they are about the length of your loaf pan. Using both hands, carefully place the babka into the prepared loaf pan, tucking the ends under slightly if needed. Cover and let proof somewhere warm for about 45-60 minutes, or until risen and puffy.
While the loaf is proofing, preheat the oven to 350°F (180°C).
Once risen, bake the babka for 40-45 minutes, until the babka is a deep, golden brown and measures 190°F (90°C) on an instant-read thermometer.
Make the cinnamon glaze: Combine the powdered sugar, milk, and cinnamon together in a small bowl until smooth and combined. If the icing is too thin, add a little more powdered sugar; if it's too thick, add more milk. Drizzle icing over warm babka. Cover leftover babka tightly and store at room temperature for up to 3 days.