Heat the oven to 375°F. Line a small rimmed baking sheet with parchment paper.
In a mixing bowl, combine the ground turkey, egg, panko, ¼ cup Parmesan, 2 tablespoons pesto, milk, salt, and black pepper. Mix gently until just combined, taking care not to compact the meat.
Shape the mixture into a compact loaf on the prepared baking sheet. Spread the remaining 2 tablespoons pesto over the top.
Bake until the center of the loaf reaches 165°F, about 40 to 50 minutes. Let it rest for 10 minutes before slicing so the juices settle.
While the meatloaf bakes, bring the chicken broth and water to a boil in a medium saucepan. Slowly whisk in the polenta. Reduce the heat to low and cook, stirring often, until thick and tender according to the package directions.
Stir the butter and ½ cup Parmesan into the cooked polenta. Spoon it onto plates and serve with slices of pesto-glazed turkey meatloaf.