Combine cream, milk, ¼ cup sugar, honey, zest, lemon juice, and salt in a medium saucepan over medium-high heat. Cook mixture, stirring frequently, until steaming and registers 175°F on an instant-read thermometer, 6–8 minutes. 

Meanwhile, whisk together egg yolks and remaining ¼ cup sugar in a large bowl until smooth and light in color, 1 minute. 

Slowly whisk 1 cup heated milk mixture into egg yolk mixture to temper. Add yolk mixture to remaining milk mixture in saucepan and cook over medium-low heat, stirring constantly, until mixture thickens and registers 185°F on an instant-read thermometer, 5 minutes. 

Off heat, stir in poppy seeds, butter extract, and vanilla extract.

Transfer custard to a large bowl and cool completely. Press a piece of plastic wrap onto surface of custard and transfer to refrigerator; chill completely, at least 4 hours.

Transfer chilled custard to bowl of ice cream machine. Churn, according to manufacturer's instructions, until mixture resembles very thick soft serve, 15–20 minutes.

Transfer ice cream to frozen baking pan and press a piece of plastic wrap onto surface; freeze until edges are firm, about 1 hour. 

Transfer ice cream to an airtight container, pressing to compress and remove air pockets. Freeze until firm, 2 hours.