Heat oven to 350°F. Grease two 8-inch round baking pans and line with parchment paper.
In a small bowl, whisk the ground flax seeds with the boiling water to form a paste. In a food processor, process the rolled oats until very fine.
In a large bowl, whisk together the flax seed paste, vegetable oil, yogurt, brown sugar, vanilla, and eggs. In a medium bowl, whisk together the ground oats, cocoa powder, salt, and baking powder. Add the dry mixture to the wet mixture and stir well to combine.
Evenly divide batter between pans and bake for about 30 minutes, until cake is firm and a tester inserted in the middle comes out clean.
Remove from oven and let cool in pans 5 minutes, then turn out onto a wire rack and cool completely. Fill and frost with the brown sugar frosting.
Frosting
In a large bowl, cream the butter and brown sugar. Add the chocolate and vanilla and beat well to combine. Add the powdered sugar and beat well until smooth. Add a little milk if frosting is too dry.