01 - In a large mixing bowl, whisk together all-purpose flour, granulated sugar, instant yeast, and fine sea salt. Add lukewarm whole milk, eggs, and vanilla extract. Mix until a shaggy dough forms. Gradually incorporate softened butter and knead until smooth and elastic, approximately 8 to 10 minutes by hand or 5 minutes using a stand mixer.
02 - Transfer dough to a lightly greased bowl. Cover and let rise in a warm environment until doubled in volume, about 1 1/2 hours.
03 - Grind poppy seeds in a spice grinder or mortar and pestle. In a small saucepan, combine ground poppy seeds, whole milk, granulated sugar, honey, and unsalted butter. Heat gently, stirring until thickened, roughly 5 minutes. Remove from heat and add lemon zest, vanilla extract, and a pinch of salt. Allow to cool completely.
04 - On a lightly floured surface, roll the dough into a 12x16-inch rectangle. Evenly spread poppyseed filling across the dough, leaving a 1/2-inch border. Roll the dough tightly from the long side to form a log.
05 - Using a sharp knife, slice the log lengthwise into two pieces. Twist the halves together with the cut sides facing upward, creating a defined braid. Place the shaped dough into a parchment-lined 9x5-inch loaf pan. Cover and allow to rise for 45 to 60 minutes until puffy.
06 - Preheat oven to 350°F. Bake babka for 35 to 40 minutes until golden and fully set. If the surface browns rapidly, tent loosely with foil in the final 10 minutes.
07 - Cool babka in the pan for 15 minutes, transfer to a wire rack. Mix powdered sugar with milk or lemon juice until pourable and drizzle over cooled babka. Slice and serve.