Lightly brush 2 baking sheets with vegetable oil. Lay the ham on top in a single layer and lightly brush with vegetable oil. Bake in the oven until crispy, 10 to 12 minutes, flipping halfway. Remove and let cool.
Once your pizza dough has thawed, divide into 3 balls. Sprinkle flour (or cornmeal for texture) on your workspace and roll each ball into an elongated "flatbread," 10 to 12 inches long and 4 inches wide. Brush both sides with olive oil.
Melt the butter in a saucepan over medium-high heat and saute the peaches with the brown sugar and some salt and pepper until caramelized, 3 to 4 minutes.
Heat a grill or large grill pan to high heat.
Grill 1 oiled crust until it has distinguished grill marks, about 2 minutes. Flip and then layer with 2 cups arugula, generous spoonfuls of the burrata, some caramelized peaches and crisp serrano. Sprinkle with salt and pepper and close the grill until the cheese is melted. Remove from the grill and repeat with the 2 remaining pizzettes.