Thinly slice 2 cups strawberries. In a blender or food processor, whirl remaining 1 quart strawberries until smooth. Press through a fine strainer into a bowl, extracting as much liquid as possible with the back of a spoon or a spatula; discard residue. Whisk 1/4 cup sugar and the lemon juice into strawberry purée.
In the blender or food processor (no need to rinse), whirl cream cheese and 1/2 cup of the strawberry purée until blended and smooth.
In a bowl, mix flour, baking soda, salt, and remaining 2 tablespoons sugar. In another bowl, whisk eggs, buttermilk, and milk just to blend. Whisk milk mixture into flour mixture just until evenly moistened.
Place a nonstick griddle or a 12-inch nonstick frying pan over medium heat (350°); when hot, coat lightly with oil and adjust heat to maintain temperature. Spoon batter in 1/2-cup portions onto griddle and cook until pancakes are browned on the bottom and edges begin to look dry, about 2 minutes; turn with a wide spatula and brown other sides, 1 1/2 to 2 minutes longer. As pancakes are cooked, transfer to baking sheets and keep warm in a 200° oven. Coat pan with more oil as necessary to pan to cook remaining pancakes.
Spread each pancake with about 2 tablespoons cream cheese mixture and roll up. Set two roll-ups on each plate, top evenly with sliced strawberries, and drizzle with a little strawberry purée; serve with remaining purée to add to taste.