Preheat oven to 350°. Butter and flour a 9 inch round cake pan.
Melt three tablespoons of butter in a large sauté pan over a medium heat. Add water and brown sugar, cook over a medium heat until brown sugar is dissolved, about three minutes. Reduce the heat to medium-low and add pears. Cook another two minutes. Set aside.
Melt four tablespoons of butter in a small saucepan over a medium heat, swirl butter until it begins to brown and smells nutty. Pour into a bowl and stick in the freezer for about 10 minutes until it becomes a consistency close to softened butter. (you may have to stir every five minutes to make sure it doesn’t freeze)
Whisk flour, salt, baking soda, and baking powder together in a medium bowl.
Cream softened butter and softened brown butter with sugar in the bottom of a stand mixer over a medium-high speed until combined, about three minutes. Reduce speed to low and add eggs one at a time, until each is incorporated. Add vanilla.
With the mixer on low, add flour and buttermilk in three batches, alternating first with the flour and then buttermilk. Mix until combined.
Arrange pears in the bottom of the greased cake pan. Pour the caramel over.
Working carefully, dollop the batter over the pear mixture and spread the batter carefully and evenly over the pears, making sure not to move the pears.
Bake for 25-30 minutes or until a wooden skewer comes out with just a few crumbs attached.