Heat your whole milk in the microwave until it reaches 115 degrees. Stir in the sugar, vanilla extract, and the yeast. Let the mixture sit and bloom for 5-10 minutes, or until it's bubbly and frothy. In a large bowl, or the bowl of a stand mixer, stir together the eggs and soft butter- it will be a bit lumpy, but that's ok!  Add the yeast mixture into the egg mixture, and stir to combine. Add in 3 cups of flour and the salt, and knead together until a dough forms, using the dough hook on a stand mixer. If the dough is super sticky and wet still, add a few more tablespoons of flour until it's pulling away from the sides of the bowl. It should be very soft still, but won't stick to your fingers. Knead the dough for 5-8 minutes, or until it's very elastic and smooth. Place the dough back in your bowl and cover. Let the dough rise until it's risen by about 1/3 in size, which takes about 30-45 minutes for me.

In a small bowl, stir together the brown sugar, cardamom, lemon zest, salt, and melted butter until combined. Grease a 12 cup muffin tin. Once the dough has risen, roll it out on a lightly floured surface into a large rectangle that is about 1/4 inch thick. Spread your filling evenly all over the dough, all the way to the edges. Slice the dough into 12 long strips that are about 1 inch thick. Fold each strip in half with the filling on the inside, and twist it into a spiral. Then, twist the spiral into itself, creating a small roll shape. Place each roll into a muffin tin cup.  Cover it in plastic and let the dough rest for about 20 minutes while you preheat your oven to 375 degrees. Remove the plastic wrap and bake the babka for 25-30 minutes, or until deep golden brown and bubbly. The filling will bubble out a little bit, but it washes off very easily so don't worry! Give the muffins a little twist and lift when they come out of the oven to ensure they don't stick to the pan.

In a small sauce pot, stir together the lemon juice, sugar, and salt, and bring the mixture to a boil over medium heat. Remove from the heat, and cool for 10 minutes. Stir in the cardamom and salt. When your babka muffins come out of the oven, immediately brush the tops generously with your syrup mixture. Cool the babka muffins for 10-15 minutes in the pan, and then place them on a cooling rack to finish cooling!