Preheat oven to 350ºF and line bottom of two, 9" round cake pans with parchment paper, greasing edges with oil or butter; set aside.
In medium bowl, combine flour, pumpkin pie spice, baking soda, baking powder, and salt; set aside.
In large bowl whisk together maple syrup and oil. Add brown sugar and pumpkin, whisking until smooth. Add in eggs and vanilla, whisking again until fully combined.
Pour half of the flour mixture into bowl of wet ingredients, stirring until almost incorporated. Add milk, vinegar and water, stir gently a couple of times, then stir in the rest of the flour mixture, being careful not to over-stir.
Evenly distribute batter between two lined pans, about 2 1/4 cups each. Bake for 15 minutes, then turn pans and bake for another 10 minutes. Cake is done when inserted toothpick in center comes out clean. Allow cake to rest in pans for 15 minutes before cooling completely on wire rack.
Meanwhile, make the frosting: In a large bowl using hand mixer or paddle attachment of stand mixer, beat together softened cream cheese with butter until smooth, add in vanilla bean seeds and powdered sugar, mixing on low speed working to medium-high speed. To thin out frosting slightly, add in 2 tablespoons of milk.
Frost the cake: Spread a couple of tablespoons of frosting at base of cake plate or cake stand (this will help hold the cake in place). Place one cake upside down onto the frosting. Spread an even layer of frosting on top of layer, about 1/2″ thick. Top other cake onto frosting (you can place layer upside down or bottom side down and shave off any extra round height from top if you’d like) and frost rest of cake. If you don't have an off-set spatula for frosting cakes, they are great. Chill cake for at least 45 minutes before slicing into. Cake can be made 1 day in advance, store covered in refrigerator.