1. Preheat oven to 350 degrees. Butter an 8-by-2-inch round cake pan; line with a parchment paper round and butter again.

2. Bring 4 cups water to a boil in a medium saucepan. Add orange and boil for 5 minutes. Drain, refill saucepan with 4 cups water and boil again. Add orange and boil for 5 minutes.

3. Meanwhile, in a medium bowl, sift together flour, 1/2 cup granulated sugar, baking powder, baking soda, and salt; set aside.

4. Remove orange from boiling water and add remaining 1 cup granulated sugar to boiling water. Cook, stirring, until sugar is dissolved. Return orange to boiling water and cook, until orange is softened, about 5 minutes. Drain and transfer orange to the bowl of a food processor along with 1/4 cup yogurt, orange zest, egg, honey, and vanilla. Process until well combined.

5. Add flour mixture to bowl of food processor; process until combined. Transfer to prepared cake pan and bake until golden brown and a cake tester inserted into the center of the cake comes out clean, 25 to 30 minutes.

6. Meanwhile, in a small saucepan, whisk together lemon juice, confectioners' sugar, and remaining 1/4 cup of yogurt. Bring to a boil over medium-high heat. Let boil until glaze is thickened, about 3 minutes.

7. Remove cake from oven and invert onto a wire rack set over parchment paper. Remove parchment paper round; pour glaze over cake. Let stand until glaze is absorbed. Serve cake with sliced baked apples.