Bloom the yeast: Stir warm milk with 1 tbsp dough sugar and the yeast. Let stand until foamy, 5–10 minutes.
Mix the dough: Whisk in beaten eggs and vanilla. Add softened butter and mix until slightly combined.
Knead: Add flour, remaining dough sugar, and kosher salt. Knead until smooth and elastic, 8–10 minutes.
First rise: Place dough in a greased bowl, cover, and let rise until doubled, about 1 to 1½ hours.
Cook the filling: Melt butter in a skillet. Add apples, brown sugar, cinnamon, and fine salt. Cook until tender, 6–8 minutes.
Thicken and cool: Stir in lemon juice and cornstarch. Cook 1 minute until glossy. Cool completely.
Prep the pan: Line a 9×5-inch loaf pan with parchment, leaving overhang. Lightly grease the sides.
Shape: Roll dough to a 10×16-inch rectangle. Spread cooled filling, leaving a ½-inch border. Roll up tightly from a long side.
Twist: Split the log lengthwise. Twist the two strands cut-sides up. Snugly fit into the pan. Sprinkle coarse sugar if using.
Second rise: Cover and let puff until slightly risen, 30–45 minutes. Preheat oven to 350°F.
Bake: Bake 35–40 minutes until deep golden and an instant-read thermometer reads about 195°F in the center.
Make syrup: Simmer water and sugar until dissolved. Brush over hot babka. Cool in pan 20 minutes, then lift out to finish cooling.