In a saucepan, whisk egg yolks and milk until combined. While continuing to stir, heat over medium heat until just before simmering. Remove from heat and strain through a fine mesh strainer into a bowl. Cover with plastic wrap, pressing wrap on the surface of the custard. Place in the coldest part of the refrigerator for 2-3 hours, or until well chilled.
In another bowl, mix the lemon zest, juice, honey, and extracts with a whisk until well blended. Refrigerate until ready to use.
Once all the ingredients are well chilled, combine lemon syrup, custard, heavy cream, and salt in a mixing bowl or large measuring cup with a pour spout.
Mix in an ice cream maker according to manufacturer's directions; most take between 20-30 minutes.
Serve immediately for soft serve ice cream or freeze in an air tight container for 2 hours for a firmer consistency.