Heat oven to 375°F. Slice 1½” off top ends of pumpkins; discard. Scoop out seeds and stringy pulp; discard pulp and reserve seeds for another use, if desired. Slice ¼” off bottom ends of pumpkins to help stand upright, if needed. Place pumpkins side-by-side in 9”x13” baking dish. Brush outside skin with 2 tsp. olive oil; set aside.
Heat remaining olive oil in medium skillet over medium-high heat. Add ham; cook, stirring occasionally, until light golden brown on all sides, about 4 minutes. Add onions to pan; cook, stirring occasionally, until onions soften, about 7 minutes. Stir in spinach and Adobo; cook until warmed through and well combined, about 3 minutes. Pour in evaporated milk; bring to boil. Stir in breadcrumbs, cooking until thickened. Remove pan from heat; stir in cheese until melted.
Evenly divide spinach mixture among cavities of pumpkins, mounding filling on top. Sprinkle remaining breadcrumbs over filling. Add 1” boiling water to bottom of baking dish; cover loosely with foil. Bake until pumpkins are tender (thin knife inserted into side will show little resistance), about 1 hour. Remove foil and bake until browned and bubbly on top, about 30 minutes more. Let cool at least 10 minutes, and up to 30 minutes, until set. Cut into wedges and serve (note skin is not edible).
This Calabaza Rellena-Stuffed Pumpkin recipe can be made the day before the big holiday dinner since the sturdy pumpkins hold up well to refrigeration and reheating. To do, remove the finished pumpkins from the oven (step 3), let cool completely (do not slice into wedges). Then cover well and refrigerate for up to 2 days. To reheat, tent with foil and heat in 350°F oven until warmed through, about 30 minutes, slice and serve.