Heat the olive oil in a large pot or cast iron skillet over medium heat. Add onion, shallot, and garlic. Cook, stirring, until the onions sweat and soften--about five minutes. Add in salt, pepper, and Greek seasoning. Stir and cook for about a minute to bloom the spices. Add in ground beef (or lamb) and brown, stirring and breaking apart large chunks. Add tomato paste and red wine. Stir to combine, and then cook, stirring occasionally, until the liquid from the wine reduces away. Then add chicken broth. Again stir to combine, and then cook, stirring occasionally, until liquid reduces away. Remove from heat, transfer to bowl, and allow to cool for 10 minutes or so. Once the meat mixture has cooled, add the feta cheese and egg and mix well. (The meat mixture just needs to be cool enough not to cook the egg or melt the feta.)

Preheat oven to 350 degrees. Melt butter, and drizzle a little butter into the bottom of a 9x13 casserole dish. Open the packet of phyllo dough and spread the sheets out. Working quickly, lay a single sheet into the pan, brush with butter, and follow with another sheet, then more butter. Alternate in this way until eight to ten sheets have been used. If possible, lay sheets so they drape a little over the edge of the pan, working around the pan twice so that a "nest" of sheets is made with dough that can be folded over the top after the filling is added. Ideally, there would be no holes where the pan shows through, and no way for filling or juices to escape the phyllo "nest." Don't worry if sheets fall apart, or the stack of layers is messy. It won't matter in the end!

Add filling, spreading evenly into the phyllo "nest." Fold the outer layers of phyllo one by one onto the filling, brushing with butter wherever one layer goes over the top of another. Then add additional layers, brushing with butter, until another 8-10 layers of dough are on top of the filling. Brush the final layer with butter. 

Take a sharp knife and "score" the top layers of phyllo, cutting down to the filling but not all the way through to the bottom layers. The idea here is to pre-cut the phyllo top into the size pieces you wish to serve, so the pie can be served in a pretty way, without shattering the crunchy top.

Bake 25-30 minutes, or until golden brown. Remove from oven, let rest five minutes, and then cut pieces along the scored lines from above. Plate and serve hot!