Melt 2 tablespoons butter in a Dutch oven or large saute pan over medium-high heat. Arrange leeks with the cut sides facing down in the pan and sear until browned. Pour chicken broth and cover the Dutch oven. Transfer to the oven and continue to cook for 1 hour and 30 minutes or until leeks are deeply caramelized. If using a saute pan, transfer the seared leeks to a baking dish, add chicken stock, and cover tightly with aluminum foil. Transfer dish to the oven and braise.
Remove leeks from the oven and transfer to a blender. Add heavy cream, parmesan, lemon juice and zest, and 2 tablespoons butter. Puree until smooth. Season with salt and pepper and set aside.
Melt remaining 1 tablespoon in a large saute pan. Add canned corn and saute until warm, about 5 minutes. Remove from heat and stir corn into leek filling.
Heat mini phyllo cups according to directions on package.
Arrange phyllo cups on a serving platter and fill with braised leek and corn filling. Drizzle reduced balsamic vinegar on top and sprinkle with chopped chives. Serve immediately.
Notes
The braised leeks must be hot in order to emulsify with the heavy cream and butter properly. If it looks like the filling is separating, heat it over medium heat, stirring frequently. If you have extra filling left over, you can use it to make pasta sauce. Thin out the sauce using the pasta water from when you cooked the pasta and toss it together with the pasta.