Preheat oven to 300 degrees F (150 degrees C). Line a 9x13-inch baking pan with parchment paper; coat with cooking spray.

Mix 3/4 cup white sugar, egg yolks, egg, vanilla extract, 1 teaspoon cardamom, and 1/2 teaspoon salt together in the bowl of a stand mixer fitted with the paddle attachment; mix on low speed until well combined.

Heat cream in a small saucepan over medium heat, stirring often, until it begins to bubble, about 5 minutes; remove from heat. Add cream to the egg mixture slowly, mixing on low speed. Pour custard into the prepared baking dish.

Set the baking pan into a roasting pan; carefully pour enough boiling water into the roasting pan to come halfway up the sides of the baking pan.

Bake in the preheated oven until the custard is set when gently shaken, 45 to 50 minutes.

Remove baking pan from roasting pan. Cool to room temperature, about 30 minutes. Freeze until firm, at least 1 hour.

Melt butter in a skillet over medium heat. Add brown sugar, salt, 1/4 teaspoon cardamom, and cinnamon; stir to combine. Add frozen pumpkin and simmer, stirring often, until reduced, about 8 minutes. Cool candied pumpkin to room temperature, about 10 minutes.