Preheat oven to 350 F. Spray the inside of a bundt cake pan with baking spray.
Cream butter and brown sugar together in the bowl of a stand mixer with paddle attachment fitted, about 3 minutes (or in a large bowl using a hand mixer). Add vanilla extract and eggs, one at a time, scraping down the sides between additions.
In a separate bowl, combine pastry flour, baking powder, baking soda, and salt. Whisk to combine.
Add half of the flour mixture to the butter and sugar, then mix until flour is completely mixed in. Add Greek yogurt and mix to combine. Then add remaining flour, scraping down sides after adding.
Pour batter into prepared bundt cake pan, then smooth the top with the back of a spatula. Bake for 35 minutes, or until cake is golden brown on the outside and springs back slightly when you push on it.
Let cool in pan for at least 15 minutes, then invert onto a wire rack to cool completely before adding glaze.