This savory pull-apart bread is as messy, ooey and gooey as the sweet version. The everything bagel seasoning of sesame seed, garlic powder, sea salt, onion and poppy seed makes for a crunchy outer shell to contrast the rich cheesy center.
For the brioche: In the bowl of a stand mixer fitted with the dough hook, whisk the yeast and sugar together with the warm milk. Allow to sit for 5 to 7 minutes in a warm, draft-free place while the yeast blooms and forms a foamy raft on the surface. When this happens, your yeast is alive and ready to go.
Whisk the eggs into the yeast mixture until thoroughly combined. Add the all-purpose flour, bread flour and salt. Start the mixer on low speed and mix for 2 minutes, until the dough begins to come together into a shaggy mass. Increase the speed to medium and mix the dough for 2 more minutes more to activate the gluten.
Add the softened butter 1 tablespoon at a time, allowing each addition to completely incorporate before adding the next. The dough will get very sticky. Continue mixing until the dough is shiny and smooth, 3 minutes.
Transfer the dough to a well-greased large bowl, and cover with plastic wrap or a kitchen towel. Allow the dough to rise in a warm, draft-free place for 1 hour.
After 1 hour, remove the plastic and scoop the dough onto a lightly floured surface. Punch down the dough by pressing it firmly with your hands until it deflates back to its original size. Spread it out slightly and fold the dough into thirds like a business letter, bringing the top half over the center and the bottom half up and over the center. Place the dough back into the greased bowl, seam side down. Cover with plastic wrap and allow to chill in refrigerator for 6 hours or overnight.
Remove the chilled dough and punch down once more. Divide the dough into 4 equal pieces, then divide each piece into 6 smaller pieces for a total of 24. Fold each piece of dough in towards itself and shape into a tight ball.
For the filling: Mix the Parmesan and bagel seasoning together in a medium bowl. Put the Gruyere in a separate bowl.
Grease a 12-cup bundt pan with a layer of butter or shortening. Sprinkle with a layer of the Parmesan seasoning mixture, about 1/2 cup, followed by a layer of Gruyere, about 1/2 cup.
Dip each dough ball in the melted butter and roll in the Parmesan mixture to coat completely. Place each dipped dough ball in the prepared pan. Once you have a layer in the bundt pan, sprinkle with another 1/2 cup Gruyere, then continue topping with another layer of dipped brioche balls. Continue this process until you have 3 layers of dough balls. Top with any remaining Gruyere and Parmesan seasoning mixture.
Cover the pan loosely with plastic wrap or a kitchen towel and allow to proof in a warm place until puffy and doubled in size, about 40 minutes. Preheat the oven to 350 degrees F.
Bake until golden brown, 35 minutes.
For the dipping sauce: Add the cream cheese to the bowl of a stand mixer fitted with the paddle attachment. Beat the cream cheese on medium-low speed until completely smooth. Add the sour cream, lemon zest, parsley, dill and a pinch of salt. Mix until smooth. Garnish with scallion greens.
Turn the monkey bread out onto a platter while still warm and serve with the dipping sauce.