Pumpkin Bread Pudding with Homemade Caramel Sauce

Bread Pudding:
14 oz loaf stale French bread, cubed
15 oz pumpkin puree
4 eggs
1/2 cup granulated sugar
1/4 cup light brown sugar
1 tsp cinnamon
1/2 tsp nutmeg
1/4 tsp ground ginger
1/2 tsp salt
1 tsp vanilla extract
1/2 cup (1 stick) unsalted butter, melted
1 cup milk
Caramel Sauce:
1 cup granulated sugar
1/2 cup water
1 Tbsp unsalted butter
1/2 cup milk, room temperature
1/8 tsp salt