In a medium saucepan over medium heat, combine the heavy cream, whole milk, instant espresso powder, and 1/4 cup of the granulated sugar. Heat until steaming and the espresso is fully dissolved, but do not let it boil. Remove from heat.
In a medium heatproof bowl, vigorously whisk the egg yolks with the remaining 1/4 cup of granulated sugar until the mixture is thick, smooth, and pale yellow.
While whisking the egg yolk mixture constantly, slowly drizzle in about 1 cup of the warm cream mixture. This gently raises the temperature of the eggs so they do not scramble.
Pour the tempered egg mixture back into the saucepan with the remaining warm cream. Place over low heat and stir constantly with a heatproof spatula or wooden spoon. Cook until the custard slightly thickens and cleanly coats the back of a spoon (about 8 to 10 minutes). Do not let it boil.
Remove from heat and stir in the vanilla extract and sea salt. Strain the custard through a fine-mesh sieve into a clean bowl. Press plastic wrap directly onto the surface of the custard to prevent a skin from forming. Refrigerate for at least 4 hours, or overnight, until completely cold.
Pour the completely chilled custard base into your ice cream maker and churn according to the manufacturer's instructions until it reaches a thick, soft-serve consistency (roughly 20 to 25 minutes).
In the final 2 minutes of churning, gently drop in the crushed meringues and chopped dark chocolate shards so they are evenly distributed.
In a small bowl, dissolve the brown sugar into the hot shot of espresso to create a thick syrup; let it cool slightly. As you transfer the churned ice cream into a freezer-safe container, alternate layers of ice cream with a drizzle of the espresso syrup, swirling gently with a butter knife.
Cover tightly and freeze for 4 to 6 hours until firm and scoopable. Let it sit on the counter for 5 to 10 minutes before serving for the ultimate creamy texture.