Combine milk and heavy cream in a saucepan over medium heat. Add fresh basil leaves and warm for 15-20 minutes, stirring occasionally.

Remove from heat and strain the mixture through a fine-mesh sieve into a bowl, discarding basil leaves.

Stir in sugar and lemon juice to the cooled mixture, adjusting sweetness to taste.

Cover and chill in the refrigerator for at least 2 hours until thoroughly cold.

Pour into an ice cream maker and churn until soft-serve consistency (about 20-25 minutes). Alternatively, freeze in a shallow dish, stirring every 30 minutes.

Transfer to an airtight container, smooth the top, cover to avoid ice crystals, and freeze for at least 4 hours.

Serve in dishes or cones, garnished with fresh berries or crushed shortbread cookies.