Preheat the oven to 375℉. Line a loaf pan with a parchment paper sling, and spray with cooking oil spray. Set aside.
If using the butter, In a medium saucepan over medium heat, add the butter. Allow the butter to melt and simmer, swirling the pan as it begins to get foamy, cooking for about 4-5 minutes. Once it begins to look golden and smell nutty, remove from the heat. Set aside to cool (to help speed up the process, I typically pour the browned butter in a bowl and place in the fridge).
Meanwhile, in a stand mixer fitted with the whisk attachment, or in a medium mixing bowl using electric beaters, beat the eggs at medium speed for 2-3 minutes until very frothy and pale yellow. It’s important you beat the eggs for the full time so that they take on air, which will help the loaf rise.
When the brown butter is cool to the touch, add it (or the avocado oil, ghee or coconut oil*) to the eggs along with the coconut sugar, maple syrup, pumpkin puree, vanilla, lemon zest and juice, and the baking soda. Whisk well to combine. Let it set for about two minutes prior to adding the dry ingredients.
Add the almond flour, tapioca flour, ground flax, pumpkin pie spice, nutmeg, and salt. Mix on low until just combined.
Pour the batter into the prepared loaf pan. Transfer the loaf to the middle rack of the oven, and bake for 55-60 minutes, or until a knife can be inserted into the center of the loaf and it comes out clean. If the top of your loaf starts to become overly browned, tent with foil and continue cooking until the bread is cooked through.
While the loaf is cooking, make the icing. In a medium bowl, using a hand mixer, add the cream cheese, almond milk, and maple syrup. Beat on low until smooth and creamy. Slowly add in the powdered sugar, vanilla, pumpkin pie spice, and a pinch of salt. Beat for one minute until well combined. Transfer to the fridge until you’re ready to ice the loaf.
Use the parchment sling and remove the bread from the pan and transfer to a baking sheet fitted with a wire rack. Using a butter knife or offset spatula, smooth the frosting over the surface of the loaf, allowing some to drip down the sides. Transfer to the fridge to allow the glaze to set a bit. Slice and enjoy!