Combine raspberries, 1/2 cup sugar and lemon juice in a saucepan over low heat. Cook and stir until sugar dissolves, about 5 minutes.
Press raspberry mixture through a strainer into a bowl. Discard seeds. Refrigerate raspberry puree until cold.
Combine remaining sugar, cream and milk in a saucepan over low heat. Cook and stir until sugar dissolves, about 5 minutes. Stir in vanilla bean paste. Pour into a bowl and refrigerate until cool, about 1 hour.
Pour cream mixture into an ice cream churner and freeze to soft-serve consistency according to manufacturer's instructions, 15 to 20 minutes. Pour into a freezer-safe container; swirl in the raspberry mixture. Freeze until firm, 8 hours to overnight.