To make the dough: put the milk into a large bowl (or the bowl of a stand mixer) and sprinkle over the yeast and a pinch of the sugar. Let stand for a few minutes, until it gets foamy. (If it does nothing, toss it and get some fresh yeast.) Add about half the flour along with the remaining sugar, eggs, vanilla and salt and stir or beat with a dough hook until well combined. Add the remaining flour and stir or beat, adding the butter a few pieces at a time, until you have a sticky, smooth dough. (It should be very soft and tacky, but as sticky as batter - it will smooth out and be easier to handle as it rises.) Shape into a ball, place in the bowl and cover with a tea towel for 2 hours.
To make the filling, stir together the sugar and cinnamon in a small bowl. Stir in the butter and honey until smooth.
Line two 8x4-inch loaf pans with parchment. Punch the dough down and divide it in half. Roll out each piece on a lightly floured surface into about a 10x12-inch square/rectangle. Spread each piece with half the cinnamon mixture. Starting at a long side, roll up jelly roll style. Cut the rolls in half lengthwise, then lay them side to side and pinch them together at the top to join, then weave the pieces back and forth over each other, like braiding but with only two pieces. Tuck into the baking pan, tucking in the edges. Don’t worry about it being perfect.
Cover and let rise for another hour or two. When you’re ready to bake, preheat the oven to 350F, brush the tops of the loaves with beaten egg, and bake for about 45 minutes, or until deep golden and the middle is springy to the touch. (Some recipes say the bottoms should sound hollow when tapped, but I find the dough is too dense and loaded with filling for this to be an accurate gauge.) Try to let them cool almost completely before slicing - but babka is pretty fabulous while it’s still warm.
Makes 2 loaves.