Preheat the oven to 325°F. Grease and flour the bundt pan.
Add the dry ingredients to a large bowl: all-purpose flour, cake flour, matcha powder, baking soda, and table salt. Whisk together for at least 30 seconds.
Fit your stand mixer with the paddle attachment. A hand mixer and beaters are also fine. Add butter, sugar, and honey to the bowl of the mixer. Beat on medium speed (4-5 on a KitchenAid) for 5 minutes, stopping to scrape the bowl once halfway through and once at the end.
Set the mixer to medium-low speed (2-3 on a KitchenAid). While the mixer is running, add one egg to the bowl and mix until completely incorporated into the buttercream mixture. Repeat this step until all 5 eggs are incorporated. Scrape the bowl. Add the vanilla extract. Beat the mixture just faster than medium speed (6 on a KitchenAid) for at least 5 minutes. This is a great time to use for cleanup.
Stop the mixer and scrape the bowl to ensure all the batter is incorporated.
Starting with the dry mixture, alternate adding the flour and remaining liquids to the mixer bowl. For all five additions, you'll add, mix on low (2-3) until mostly combined, stop the mixer, and repeat. First, ⅓ of the dry mixture. Second, the almond milk. Third, ⅓ of the dry mixture. Fourth, the Greek yogurt. Lastly, mix in the remaining dry ingredients. Scrape the bowl again. Mix everything one last time on medium-low speed (3-4 on KitchenAid) for 30 seconds.
Pour the batter into your greased and floured bundt pan. Smooth the top of the batter if needed. Place the pan in the oven and bake the cake for 85 minutes or until the top has risen, fluffy, browned, and broken at the center.
Turn the oven off and remove the bundt pan from the oven. Allow it to cool for 10-15 minutes on the countertop or a trivet. Flip the cake onto a cake plate and allow it to cool uncovered for about 4 hours at room temperature, or until the cake is cool to the touch.