Step 1

Preheat broiler to high with oven rack 6 inches from heat. Cut kernels from corn cobs to equal about 2 cups; reserve cobs. Toss together kernels, canola oil, and 3/4 teaspoon salt on a rimmed baking sheet until kernels are well coated. Spread into a single layer, and broil in preheated oven until beginning to char, 4 to 6 minutes. Set aside.

Step 2

Using the back of a knife, scrape milk from corn cobs into a small saucepan. Add corn cobs and 1 cup whole milk. Bring to a simmer over medium-low, and cook, stirring occasionally, 5 minutes. Remove from heat. Let stand at room temperature 30 minutes.

Step 3

Pour milk mixture through a fine wire-mesh strainer into a bowl. Scrape cooked cobs with a spoon to release milk; discard cobs and other solids. Add 1 1/2 cups heavy cream and 1/4 teaspoon salt to bowl. Transfer milk mixture to saucepan, and bring to a simmer over medium-low. Meanwhile, whisk together egg yolks, 1/2 cup sugar, and 1/2 teaspoon vanilla in a medium bowl. Gradually add two-thirds of the simmering milk mixture to egg yolk mixture, whisking constantly, until incorporated.

Step 4

Stir egg yolk–milk mixture into remaining milk mixture in saucepan. Reduce heat to low, and cook, stirring constantly, until mixture thickens and coats the back of a spoon, about 12 minutes. Transfer crème anglaise to a medium bowl, and stir in lime zest. Press plastic wrap directly on surface, and cool completely, about 2 hours. Set aside. (Cooled crème anglaise can be chilled up to 1 day. To reheat, transfer to a small saucepan, and heat over low, stirring constantly, until just warm, 2 to 3 minutes.)

Step 5

Preheat oven to 325°F. Whisk together eggs, bourbon, remaining 1 cup heavy cream, remaining 1 cup milk, remaining 1/4 cup sugar, remaining 1 teaspoon vanilla, and remaining 1/4 teaspoon salt in a medium bowl. Toss together croissants, French bread, and roasted corn in a large bowl. Add bourbon-cream mixture to bread mixture, and stir to combine. Let stand at room temperature 30 to 45 minutes, stirring occasionally.

Step 6

Pour bread mixture into a lightly greased 9-inch square baking pan, and spread in an even layer. Bake at 325°F until set and lightly browned, 35 to 40 minutes. Transfer pan to a wire rack, and let cool 30 minutes before slicing. Cut bread pudding into squares; serve with crème anglaise, and garnish with caramel popcorn.