Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Whisk together the flour, salt, baking powder, baking soda and cinnamon in a small bowl. Whisk together the eggs, milk, olive oil, sugar, 1 teaspoon vanilla, lemon juice and zest in a large bowl. Pour the dry ingredients into the wet ingredients and whisk to combine. Set aside.
Place a shallow enameled Dutch oven over direct heat. Add the butter and melt. Continue to toast the butter until it becomes brown and nutty, 1 to 2 minutes. Swirl the butter up the sides of the Dutch oven as you are toasting it to prevent the cake from sticking.
After the butter has browned, whisk in the brown sugar and allow it to melt. The mixture will resemble wet sand and won't get completely smooth.
Take the Dutch oven off the heat and shingle the lemon slices in, covering the whole surface. Keep in mind that this will be the top of your cake. Pour the batter into the Dutch oven and put it back over indirect heat. Close the grill and bake until a toothpick inserted into the center of the cake comes out dry, 50 minutes to 1 hour. Remove the cake from the grill and allow it to cool for 5 minutes. Invert the cake onto a platter and continue to cool until ready to serve.
Meanwhile, place a 2-cup glass jar with a lid in the freezer for 20 minutes. When the jar is cold, pour in the cream and add the sugar and the remaining 1 teaspoon vanilla. Vigorously shake until you end up with soft whipped cream, about 5 minutes.
Cut the cake into 8 slices and serve with a dollop of whipped cream on top.
(Alternatively, you can bake the cake in a 350 degrees F oven for 50 minutes.)