Pre-heat the broiler.
For the pesto, combine all the ingredients except the EVOO in a food processor. Add about 1/4 cup EVOO and pulse into a thick sauce. Stream in a few extra tablespoons of EVOO, if necessary. Transfer the pesto to a serving bowl; reserve at room temperature.
For the burgers, mix the meat with salt, pepper, Worcestershire sauce, garlic, coriander and poultry seasoning. Form four patties. In a skillet, heat 2 tablespoons EVOO, two turns of the pan, over medium-high heat. Cook the patties for 12 minutes total, turning once.
In a small saucepot, melt the butter with the garlic. Brush the garlic butter over the toasted buns and top with cheese and pepper. Broil until golden and the cheese is melted, about 2-3 minutes.
Serve the patties on rolls slathered with pesto and top with a lettuce leaf.