Pour 1 tablespoon olive oil into a skillet, swirling to coat. Cook and stir capicola over medium heat until lightly browned and crisp, 3 to 5 minutes. Stir in garlic clove; remove skillet from heat. Transfer capicola and garlic to a bowl, reserving grease in the skillet.
Pour remaining 1 tablespoon olive oil into the skillet. Cook and stir shiitake mushrooms until tender, about 5 minutes. Remove from heat.
Preheat oven to 425 degrees F (220 degrees C).
Combine 1 tablespoon minced garlic and sesame oil in a small saucepan over medium-low heat; cook and stir until fragrant, about 2 minutes. Remove from heat. Stir soy sauce, oyster sauce, chili-garlic sauce, water, sugar, and pepper into the garlic.
Spread garlic sauce over the pizza crust. Top with capicola, giardiniera, and shiitake mushrooms. Cover capicola and mushrooms with smoked Gouda and Manchego cheeses.
Bake in the preheated oven until crust is golden and cheeses are melted, 8 to 12 minutes.