Heat oven to 375ºF. Add oil to a large skillet set on medium heat. When oil is shimmering, add the shallots, garlic and a pinch of kosher salt. Cook until just wilted, 2 to 3 minutes. Add chard and cook until wilted, 4 to 5 minutes. Turn off heat and set aside to cool.
In a large bowl, combine ricotta, feta, shredded chicken, mozzarella, eggs, scallions, dill and pepper. Stir in the cooled chard mixture.
Brush a 12-cup Bundt pan very liberally with some of the melted butter. Drape 2 sheets of phyllo on top of Bundt pan, poke through the dough to fit it over the center tube, and push phyllo into the bottom of the pan to line it. Drizzle about 1 tablespoon of butter over the phyllo. Do this with another 2 sheets placed perpendicular to the first one. Drizzle a little more butter after every few layers. Continue adding phyllo sheets in this crisscross manner, leaving any leftover butter for the top, until all sheets are used. Edges of phyllo should hang over edges of pan.
Add ricotta filling into pan. Fold edges of phyllo over filling. Using a long, sharp knife, poke about 2 dozen holes in dough that reach all the way to bottom of pan. Slowly pour the remaining melted butter all over torte; some butter will seep through holes and some will remain on top of dough.
Place pan on a baking sheet and bake for 60-70 minutes, until torte is puffy and golden brown. Allow torte to cool in bundt pan for 1 to 2 hours before inverting onto a platter and slicing. Serve warm or room temperature.