Combine the heavy cream and milk in a saucepan and heat until steaming but not boiling. Remove from the heat, add the Earl Grey tea bags, cover, and steep for 10 minutes.

Gently press the tea bags against the side of the pan, then remove them without aggressively squeezing.

Whisk the egg yolks, granulated sugar, and corn syrup in a bowl until smooth. Slowly whisk in about 1 cup of the warm Earl Grey cream, then pour the mixture back into the saucepan.

Cook over medium-low heat, stirring constantly, until the custard reaches 170 to 175°F and lightly coats the back of a spoon.

Strain the custard through a fine-mesh sieve, then stir in the vanilla and salt. Chill completely, preferably overnight.

Churn the chilled custard according to your ice cream maker's instructions. Spread the freshly churned ice cream about 3/4 inch thick in a parchment-lined 8-inch square pan and freeze until completely firm, at least 4 hours.

Heat the oven to 350°F and line two baking sheets with parchment paper.

Beat the softened butter, brown sugar, and granulated sugar until creamy. Beat in the egg, egg yolk, and vanilla.

Add the flour, baking soda, and salt and mix just until no dry flour remains.

Divide the dough into 16 equal portions, about 2 tablespoons each. Roll into balls, gently flatten, and arrange on the prepared baking sheets.

Bake for 8 to 10 minutes, until the edges are set but the centers still look slightly underdone. Cool on the baking sheets for 5 minutes, then transfer to a rack and cool completely.

Use a round cutter approximately the same diameter as the cookies to cut 8 thick rounds from the frozen Earl Grey ice cream.

Sandwich each ice cream round between two cookies and press very gently. If using the chocolate, roll the exposed ice cream edges in the finely chopped chocolate.

Wrap the sandwiches individually and freeze for at least 1 to 2 hours before serving. Let stand at room temperature for 3 to 5 minutes before eating.