Best enclosure style. Need advice

[Kjskjs]

Trying to decide best enclosure for pizza oven that still looks good and matches our new patio/fireplace. We are in the mid-Atlantic area with rain and freezing temps. I know a full roof where people could stand underneath is the perfect solution but it won’t work for us because of aesthetics. Next best is probably a dog house style like in the first photo (but with more overhang), so we’re thinking of using limestone veneer to match fireplace, and have a metal roof. With mortar and stone being porous, what steps do I need to take to minimize water penetration? The plan is to build the inner igloo fully with stucco covering, and waterproof sealer, and then build the box enclosure around the igloo. But what about the voids between the igloo and box? And should I vent the box like home construction with soffit and ridge vents? Should I use cement board to veneer the stone, or something better? Appreciate any details you can give me!