Preheat the oven to 375℉. Line a 9x5-inch loaf pan with a parchment paper sling and spray with cooking oil. Set aside.
Melt the butter in a medium saucepan set over medium heat, swirling the pan as it begins to get foamy. Cook for 4 to 5 minutes. Once it begins to look golden and smell nutty, remove from the heat. Set aside to cool.
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 to 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 to the whisked eggs along with coconut sugar, maple syrup, pumpkin purée, vanilla, lemon zest and juice, and baking soda. Whisk well to combine. Let the mixture rest for about 2 minutes before 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 to 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, add the cream cheese, almond milk, and maple syrup. Use an electric hand mixer to beat on low until smooth and creamy. Slowly add in the powdered sugar, vanilla extract, pumpkin pie spice, and salt. Beat for 1 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. Let cool completely. 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!