Place the butter into a pot and cook, stirring, until it browns. Remove from the heat and let cool, but not to the point that it solidifies, about 5 minutes.
Whip the eggs and egg yolks together with the granulated sugar until thick and fluffy. Sift the flour and baking powder together in a bowl, then fold into the whipped egg mixture. Add the butter and mix until incorporated. Add the sour cream, milk and vanilla seeds and mix until it comes together. Transfer to a piping bag.
Spray two 12-cup muffin tins generously with cooking spray. Arrange the sliced figs in the muffin cups (overlapping is okay). Sprinkle about 2 teaspoons brown sugar on top of each. Fill the muffin cups with the cake batter about three-quarters of the way full. Bake until golden brown and a toothpick inserted in the edges comes out clean, about 15 minutes. Allow to cool slightly before removing from the muffin cups.
For the sweet ricotta: Meanwhile, put the ricotta in a large fine-mesh sieve set over a bowl and let drain for 30 minutes. Discard the liquid. Stir in the confectioners' sugar, vanilla and salt until smooth. Reserve until ready to serve.
Serve the cakes with a dusting of confectioners' sugar and dollops of sweet ricotta. Garnish with quartered figs if using.