For the Brioche dough:

Lightly grease a large bowl with the vegetable oil and set aside. Combine the warm milk and the yeast in a large bowl and stir to combine. Let sit until foamy, about 5 minutes. Add 1/4 cup of the flour and stir to combine. Cover with plastic wrap and let rise in a warm, draft-free place until doubled in size, about 1 hour.

Using an electric mixer fitted with a dough hook, beat the eggs, 1 at a time, into the yeast mixture.

Slowly add the remaining 1 cup of flour, the sugar, and salt, and beat until well incorporated, about 5 minutes. Add the butter 2 tablespoons at a time, beating well between additions. Blend until sticky dough forms. Place in the oiled bowl, lightly turning to coat. Cover with a kitchen cloth and let rise in a warm, draft-free place until doubled in size, about 1 hour.

Preheat oven to 375 degrees F.

Turn out the brioche dough onto a lightly floured surface and punch down. Roll out the dough to a circle about 14-inches in diameter.

Fit dough into a 9-inch pie pan, leaving about 1/2-inch overhang for shrinkage. Line with parchment paper and fill with pie weights. Bake until set, about 15 minutes. Remove from the oven. Remove the parchment and beans, return to the oven and bake until lightly colored, about 5 minutes. Remove from the oven and cool on a wire rack.

For the Filling:

In a medium bowl, combine the dark rum and honey, and stir. Add the figs and marinate for 30 minutes. Drain and discard the liquid.

Reduce the oven temperature to 350 degrees F.

Combine the ricotta and cream cheese until smooth in an electric mixer. Add the whole eggs and egg yolk, 1 at a time. Add sugar, lemon juice, zest, vanilla, and cardamom and mix until combined. Drain the figs and fold into ricotta mixture. Pour mixture into prepared shell. Do not overfill the shell, as it will rise slightly when baking.

Bake until the filling is firm to the touch, about 1 hour. Cool before serving. Garnish with powdered sugar and mint. Refrigerate any leftovers.