In a medium saucepan, combine the milk and cream. Place over medium-low heat and cook, stirring occasional so a skin doesn't form. You want the temperature to reach 170°F.
While heating milk/cream mixture, whisk egg yolks in medium bowl. Gradually add 2/3 cup of sugar and xanthan gum and whisk until mixture is thick and pale yellow. Temper egg yolks by slowly pouring 1 cup hot milk mixture at a time, whisking constantly. Return egg yolk mixture to to saucepan and place over low heat. Cook, stirring frequently with a wooden spoon until the custard reaches a temperature of 185°F. Pour the mixture through a fine-mesh strainer into a clean bowl and cool to room temperature, stirring every 5 minutes. Once room temperature, cover and place in refrigerator for at least 4 hours or overnight.
In a medium saucepan, add strawberries, 1/4 cup of sugar and lemon juice. Let sit for 15 minutes.
Place saucepan over medium heat and cook until strawberries are softened completely and the syrup just starts to thicken, about 10 minutes. Remove from heat and let cool to room temperature. Transfer to airtight contain and place in refrigerator for at least 2 hours or overnight.
Add 3/4 of the strawberry mixture to a blender or food processor, saving 1/4 of it for later. Add half of the ice cream base and vanilla extract to the blender and blend until it is fully incorporated with strawberries. Mix in remaining ice cream base.
Pour mixture into ice cream maker and mix according to manufacturer's instructions. (every ice cream maker will be different. Ours takes about 20-25 mins) With 5 minutes left, add remaining strawberry mixture to ice cream machine. Once desired consistency, pour ice cream into a container and freeze in freezer for 4+ hours to get a desired consistency.