Preheat oven to 350°F. Thoroughly butter and flour a 12-cup bundt pan, tapping out excess. Grate zucchini on the large holes of a box grater, then wrap in a clean kitchen towel and squeeze firmly to remove excess moisture. You want about 1.5 cups packed, squeezed zucchini. Set aside.

In a medium bowl, whisk together flour, baking powder, baking soda, and salt until well combined.

In a large bowl with a hand mixer or in a stand mixer fitted with the paddle attachment, beat butter and sugar on medium-high speed until light and fluffy, about 3 minutes. Scrape down the bowl. Add eggs one at a time, beating well after each. Beat in lemon zest, 2 tablespoons lemon juice, and vanilla until smooth, about 30 seconds.

With mixer on low, add dry ingredients in three additions alternating with Greek yogurt in two additions, beginning and ending with dry ingredients. Mix just until combined. Fold in squeezed zucchini with a rubber spatula. The batter will be thick.

Scrape batter into prepared pan and smooth the top. Bake until a toothpick inserted in the center comes out clean and the top springs back when lightly pressed, 45 to 55 minutes. Cool in pan on a wire rack for 15 minutes, then invert onto rack and cool completely, about 1 hour.

Whisk powdered sugar with remaining 1 tablespoon lemon juice plus 1 to 2 tablespoons water or additional lemon juice until pourable but thick. Drizzle over cooled cake, letting it run down the sides. Let set for 15 minutes before slicing.