Add the milk to a medium bowl. In another medium bowl, whisk together the yolks, sugar, and salt. Add the cream to a medium saucepan and bring to a simmer over medium-high heat.
Whisking continuously, add the hot cream mixture to the yolk mixture a little at a time. Continue whisking until well-combined. Return the mixture to the saucepan and cook over medium-low heat, stirring constantly, until it is just thick enough to coat the back of a wooden spoon, 3 to 4 minutes. (If you can draw a line through the custard on the back of the spoon and it holds its shape, then the mixture is ready.)
Pour the custard mixture through a fine mesh sieve into the bowl of milk. Discard any solids. Stir in the lemon zest and the lemon juice. Cover with plastic wrap and set in the refrigerator to chill until very cold, at least 12 hours. Place an empty metal loaf pan in the freezer to chill.
Pour the chilled custard into an ice cream maker and process according to the manufacturer’s instructions. Add the meringue cookies to the ice cream during the last few minutes of churning. Transfer the ice cream to the prepared loaf pan, wrap well, and freeze until firm, at least 6 hours.