Preheat oven to 350°. Generously butter a 8½x4½" loaf pan (I used a 9x5 pan).
Whisk egg, egg yolk, ¼ cup granulated sugar in a medium bowl until smooth. Whisk in yeast mixture.
Combine salt and 2 cups flour in the bowl of an electric mixer fitted with the paddle attachment. Add egg mixture (different than the egg wash) and beat on low speed until nearly incorporated, about 30 seconds. Switch to dough hook and add 7 Tbsp. butter; beat on medium-low until butter is completely incorporated and dough is smooth and just slightly sticky, 8–10 minutes (I used a Cuisinart with dough blade and did not knead for more than 3-4 min)
Heat all ingredients in a heatproof bowl set over a saucepan of simmering water, stirring occasionally, until chocolate and butter are melted and mixture is smooth. (Alternatively, microwave on medium power 1 minute, stirring halfway through.)
Whisk brown sugar, sugar, cinnamon, and salt in medium bowl to blend. Mix in butter. Add flour and toss with a fork until moist clumps form (streusel will look slightly wet).
Whisk egg and cream in a small bowl and reserve.
Turn out chilled dough onto a clean lightly floured surface. Roll out to a 22x12" rectangle and orient so a long side is facing you. Using an offset spatula, spread chocolate filling over dough to extend to the edges.
Roll up dough away from you like a jelly roll, pulling lightly on it as you roll to maintain thickness. (Use a metal bench scraper if needed to help dough release from surface.)
Brush the bread after 2nd rise with the reserved egg wash.
Scatter the streusel to cover all surfaces. Seems like a lot but use all of it!
Bake babka, rotating pan halfway through, until golden brown, 50–60 minutes (monitor closely after 1st turn, time may vary but you want a nice browning effect for the crust).
Transfer pan to a wire rack and let babka cool in pan 15 minutes, then turn out onto rack, running a paring knife around edges to help loosen if needed. Let cool completely before slicing.