This is a burger my mother would love—not too heavy, served on toast with the sweet, mildly oniony flavor of leeks, and highlighted with lemon.
In a medium skillet with a lid, melt 2 tbsp. butter over medium heat. Add the leeks. Cook, stirring occasionally, until softened, about 5 minutes; season with salt. Add the stock. Lower heat to medium-low. Cover and cook, stirring occasionally, until the leeks are very soft, about 10 minutes. Remove the lid and cook, stirring often, until only a little liquid remains, 1 to 2 minutes more. Add the lemon juice. Remove the melted leeks from heat.
In a medium bowl, season the turkey with salt and pepper. Mix in the herbs, sour cream, garlic, and lemon zest. Form into 4 thin patties (thinner in the centers for even cooking), not thicker than 1/2 inch.
Heat a griddle or large skillet over medium-high. Add the oil, one turn of the pan. Cook the patties until cooked through, turning once, 4 to 5 minutes per side. Wipe the skillet clean.
Lightly butter 1 side of each slice of bread. Place the bread on a work surface, buttered-side down. Spread 4 slices of bread with half the Dijon. Top with the Swiss cheese, patties, American cheese, and melted leeks. Spread the remaining bread with more Dijon and place on top, buttered-side facing out.
Heat the same pan over medium. Cook the patty melts until the bread is golden and the cheese is gooey, about 3 minutes per side. Cut each patty melt from corner to corner.