How to Make It
In a large, deep skillet, heat the olive oil. Add the onion and peppers, season with salt and pepper and cook over moderate heat, stirring frequently, until softened, about 10 minutes. Add the tomatoes and cook until softened and their liquid has evaporated, about 10 minutes. Stir in the eggplant and season with salt and pepper. Cover and cook over moderate heat, stirring occasionally, until the vegetables are softened, 15 minutes. Transfer the vegetables to a bowl and let cool slightly.
Preheat the oven to 400° and lightly oil a 9-by-13-inch glass baking dish. In a medium bowl, whisk the eggs with the milk and 1/2 cup of canola oil. Lay a sheet of phyllo on a work surface and brush with the egg mixture; repeat until you have a stack of 4 sheets. Transfer the stack to the baking dish and spread one-third of the eggplant mixture on top. Repeat the layering twice more with the brushed phyllo sheets and eggplant filling, then end with a stack of brushed phyllo.
Trim the overhanging phyllo slightly and tuck it under itself. Brush any remaining egg mixture on top of the börek and bake for about 45 minutes, until deeply golden on top and sizzling. Let the börek rest for 20 minutes before cutting into squares and serving.