To prepare dough: Grease a large mixing bowl with butter and reserve.
In a bowl of an electric mixer fitted with a dough hook, combine 21/2 cups flour, salt and all but 1 tablespoon sugar. In a small bowl, combine yeast, remaining 1 tablespoon sugar and 1 tablespoon warm water and stir just until sugar and yeast have dissolved.
With mixer on low speed, pour yeast mixture, milk, egg and egg yolks into flour mixture and mix until dough is smooth, shiny and elastic, about 10 minutes, adding a little more flour as needed. Add butter, a piece at a time, until it is incorporated, then continue to mix, on low speed, for about 5 minutes, until dough is silken and rich.
Transfer dough to prepared mixing bowl, cover with plastic wrap and let rise, at room temperature, for about 2 hours. When dough has risen, press it down, cover in plastic wrap and refrigerate for 1 or 2 hours or overnight.
To prepare tapenade filling: In bowl of a food processor fitted with a metal blade, combine olives, 1 can anchovies, fennel and 1 tablespoon olive oil. Purée mixture until smooth. Taste and add more anchovies, and another tablespoon of olive oil if filling is not smooth enough.
To assemble babka: Grease bottom and sides of 2 (9-inch round) baking pans with butter. Remove dough from refrigerator and divide in half.