Rinse, hull, and slice the strawberries. Chop half of them into very small pieces and keep half of them sliced. Add the sugar, lemon juice, and vanilla extract and mix. Do this in advance so they have time to macerate, at least a few hours or overnight. The sugar, lemon juice, and vanilla will cause the strawberries to soften and release some of their juice. Before mixing into the ice cream base, you'll want to smash them a little more with a fork or puree part of them.
To make the ice cream base, separate the eggs and put the yolks in a medium-sized bowl. Add 1/4 cup of the sugar and whisk it into the egg yolks.
Get a big bowl and fill it with ice and a little water. Get a bowl just smaller than that and set it on top of the ice. You will be pouring the custard into that bowl to cool it down quickly. Set a small fine-mesh strainer over the bowl.
In a medium sized heavy bottomed sauce pan, combine the cream, milk, and remaining 1/2 cup of sugar. Heat until just below a simmer.
To temper the egg yolks before adding them to the cream mix, take about 1 cup of the hot cream/milk mix and whisk it into the egg yolks. Then add the egg yolks, whisking slowly, back into the cream/milk mix. Cook, being careful not to boil, until the mixture begins to thicken into a custard and will coat the back of a spoon. You do not want to over cook this or the eggs will start to scramble.
Pour the custard through the strainer set over the bowl on ice. The strainer will catch any bits of egg which might have over-cooked. Stir the custard until cool and then cover and refrigerate until completely cold, preferably overnight.
Freeze the ice cream according to the manufacturer's instructions and add the strawberries in at the end during the last 5 minutes of churning.