Preheat oven to 350 degrees (175 C). Line 2 baking trays with baking paper (parchment paper).

In a bowl, combine all-purpose flour, cake flour, baking powder and baking soda. Set aside.

Put brown butter into another large bowl. Add oil, honey, brown sugar, salt, vanilla and lemon juice and stir until combined. Add egg and yolk and continue whisking (with a hand whisk, no mixer needed) until mixture is smooth. Let the mixture stand for 10 minutes at this point in order to allow sugar to dissolve.

Using a spoon or spatula, stir in flour mixture until just combined, but do not overmix. Cover and refrigerate for about 2 hours.

Take 1 tablespoon of dough at a time and roll into balls. If you like, you can roll dough balls in a combination of brown and granulated sugar to give it a crust. Place dough balls 2 inches apart on lined baking sheets. Press lightly to flatten to get 2 inches (5 cm) wide cookies.

Bake 1 sheet at a time on the middle rack, until cookies are puffy and slightly brown, about 8 to 10 minutes (depending on your oven). Remove cookies from the oven, gently press down with the bottom of a glass. Place them back into the oven for 2 more minutes. Cookie centers should still be soft. Cool the cookies on the sheets for at least 5 minutes.