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