Preheat the oven to 400 degrees F. Spray a 12-cup muffin pan generously with the baking spray.
Whisk the milk, flour, vanilla, eggs and 1/2 teaspoon salt in a medium bowl until combined. Drizzle in the melted butter while whisking until combined; let the batter stand for 5 minutes. Add the baking powder, whisking to combine, and divide the batter evenly among the muffin cups (about a scant 1/4 cup batter per muffin cup). Bake the popovers until puffed and browned around the edges, rotating the pan about halfway through, 20 to 25 minutes.
For the filling:
While the popovers are baking, whisk together the broth and flour in a medium saucepan until the flour is dissolved. Place the saucepan over medium heat and cook, whisking continuously, until the mixture comes to a boil; continue to cook, whisking, until the mixture is thick enough to coat the back of a spoon, 4 to 5 minutes. Reduce the heat to low and whisk in 1 teaspoon salt, 1/4 teaspoon pepper and half of the Cheddar. Stir in the peas and carrots, green beans and then the chicken; cover to keep warm.
When the popovers are done, turn them out of the pan, using a small spatula if necessary. Arrange 2 popovers on each of 6 plates or bowls and top with the chicken filling, dividing evenly among the popovers. Sprinkle the remaining Cheddar over the warm filling. Garnish with chopped parsley if using. Serve with the iced tea.