Preheat your oven to 375°F. Dice the onion, celery, and carrots into small, even pieces. Slice the mushrooms and mince the garlic. Shred the cooked chicken and set it aside. Measure out the broth, cream, and cream cheese.
Melt the butter in a large oven safe skillet over medium heat. Add the diced onion, celery, and carrots, and cook for 5 to 6 minutes, stirring occasionally, until the onion turns translucent. Add the mushrooms and cook for another 3 minutes. Stir in the minced garlic and cook for 30 seconds until fragrant.
Pour in the chicken broth and bring it to a gentle simmer. Add the cream cheese in small pieces, stirring until it melts fully. Pour in the heavy cream and stir to combine. Sprinkle the xanthan gum evenly over the surface while whisking constantly to avoid clumps. Stir in the thyme, poultry seasoning, salt, and black pepper.
Fold the shredded chicken into the skillet. Add the diced green beans and stir everything together. Let the filling simmer for 3 to 4 minutes so the green beans start to soften and the flavors come together. Taste and adjust the seasoning, then remove from heat.
In a microwave safe bowl, combine the shredded mozzarella and the 2 tablespoons of cream cheese for the crust. Microwave in 30 second bursts, stirring between each, until fully melted and smooth. Add the almond flour, egg, and baking powder to the melted cheese mixture. Stir with a spatula until a dough forms, then knead briefly until smooth.
Place the dough between two sheets of parchment paper and roll it into a circle slightly larger than the top of your skillet. Peel off the top parchment, carefully flip the dough over the filling, and peel away the remaining paper. Gently press the edges to seal and cut a few slits in the top to let steam escape.
Transfer the skillet to the preheated oven and bake for 20 to 25 minutes, until the crust is deep golden brown and the filling is bubbling around the edges. If the crust browns too quickly, tent loosely with foil. Remove and let rest for 10 minutes before serving.