Warm milk and buttermilk in in a pot on the stove, or alternatively in your microwave until warm to the touch, but not too hot - about 95˚F. If your milk is too hot is could run the risk of killing the yeast.
In the bowl of a stand mixer fitted with the dough hook attachment, combine the warm milk and buttermilk,1 tablespoon sugar, and yeast. Whisk together to combine gently and allow to stand for 5-10 minutes or until frothy.
Meanwhile, whisk together 4 ¼ cups of the flour and salt; set aside.
Add eggs, softened butter, and remaining sugar to the milk and yeast mixture. Turn the mixer on low to begin to combine until the butter starts to break apart into pieces. Add in 1 cup of the dry ingredients, and the turn the mixer to low speed. Add in the remaining dry ingredients in two additions, allowing the flour to incorporate a bit after each addition.
Once all your flour is incorporated, set the mixer to high and allow the hook to work the dough for about 5 minutes – you want the dough to appear smooth and elastic. Remove the dough from the bowl and turn it out onto a lightly floured work surface, and knead the dough with your hands for another few minutes, and shape it into a ball.
Place the dough into a large greased bowl. Cover it with plastic wrap and set it in a warm place to rise until it has doubled in size; about 30 - 45 minutes.
Meanwhile prepare your everything bagel spice by combining sesame seeds, poppy seeds, dried onion, dried garlic, and sea salt in a small bowl. Set aside.
To assemble grease a 9 by 5 inch loaf pan with butter and preheat oven to 350˚ F.
Once dough has doubled in size, punch it down and turn it out onto a lightly floured work surface. Use a rolling pin to roll dough to a ¼ inch thick rectangle, about 9” by 12”. Spread with ¼ cup of softened butter, leaving a 1 inch border at the bottom closest to you. Spread grainy mustard on top of the butter and sprinkle the dough evenly with your prepared everything bagel spice.
Tightly roll the dough up jelly-roll style into a 12 inch log. Use a sharp knife to slice the dough log in half lengthwise. Face the two pieces cut side up and twist the two pieces, criss crossing one piece over the other until you come to the end. Pinch the ends together, and gently place babka braid into your prepared loaf pan.
Cover the pan with a dry tea towel and set in a warm place to rise again for another 20 minutes.
Meanwhile, whisk together your egg wash; set aside.
Once your babka has finished its second rise, brush the top evenly with your prepared egg wash. Bake for 45-50 minutes or until the top is golden brown. Remove from the oven and place on a wire rack to cool for 15 minutes before slicing.