Pulse ham steak and prosciutto in a food processor until very finely chopped, about 12 times. Transfer to a large bowl. Add ground pork, egg yolk, garlic, and thyme; gently mix together, then shape into four 4 1/2-in.-wide patties. Sprinkle both sides of patties with pepper.
Heat oil in a large cast-iron skillet over medium-high heat. Add patties and cook until an instant-read thermometer inserted in centers registers 140°, about 3 minutes per side, topping with cheese during last minute of cooking. Transfer patties to a platter and tent with foil.
Add onions, 1 tbsp. water, the salt, and baking soda to skillet. Stir to combine, scraping up browned bits. Cover and cook, stirring occasionally, until onions are tender and golden, about 10 minutes. Uncover and cook, stirring often, until liquid has evaporated, 1 to 2 minutes.
Arrange 2 lettuce leaves on each bottom bun half. Top with burgers and caramelized onions. Spread mustard on top halves and place on burgers.