Preheat the oven to 425ºF and line an 8x8 baking pan with parchment paper. Beat together the flaxseed meal and water and set aside for 5 minutes to gel. In a large mixing bowl, whisk together the dry ingredients. Add the milk, flax egg, pumpkin, syrup and olive oil and stir to combine. Transfer batter to the prepared baking pan. Bake for 23 - 25 minutes until golden brown and a cake tester inserted into the center comes out clean. Remove from the oven and let cool in the pan for 5 minutes. Transfer to a wire rack and allow to sit overnight.
Prepare the stuffing:
Slice the cooled and semi-dried out cornbread into small 1" cubes. Add them to a bowl and set aside.
Heat the oil in a large skillet over medium heat. Add the celery, shallots, and garlic and saute until the garlic is fragrant. Add the mushrooms, herbs and season with salt and pepper. Continue to cook until the mushrooms are completely soft, about 6 - 7 minutes.
Add the vegetables to the bowl with the cornbread, along with the milk and broth. Stir to combine. Transfer the contents of the bowl to a 9x13 baking dish. Spread the stuffing evenly out in the dish then pop back in the oven and cook for about 15 - 20 minutes until the top is golden brown.