Croissant Bread Pudding

Name: Croissant Bread Pudding
Ingredients: Softened butter or vegetable oil, for greasing the pan; 6 to 8 medium croissants (360 grams); 1 cup whole milk; 1 cup heavy cream; 1 lightly packed cup/200 grams light or dark brown sugar; 3 large eggs, at room temperature; 1 teaspoon vanilla extract; 1/3 cup/40 grams chopped raw pecans; 1/2 cup/113 grams unsalted butter (8 tablespoons); 1 large egg, at room temperature and lightly whisked; 1/4 cup light or dark rum; Ice cream (optional), for serving