Set the ice cream out at room temperature to soften for about 30 minutes. Meanwhile, use a fork to mix together the graham cracker crumbs, melted butter and 1/4 cup sugar in a bowl. Using your fingers, press the mixture into the bottom and up the sides of a 9-inch springform pan, then press all over with the flat bottom of a glass to get the crust really well pressed together and compact.
When the ice cream has softened, beat it in a stand mixer with the paddle attachment (or by hand in a bowl with a wooden spoon) until soft and creamy but not melted. Break the cheesecake into pieces and beat or fold into the ice cream. Scrape the ice cream mixture into the prepared springform pan and smooth the top. Freeze until set, at least 4 hours or overnight.
Combine the strawberries, the remaining 2 tablespoons sugar and the lemon juice in a nonreactive saucepan and warm over medium heat just until the strawberries begin to break down and give off their juices, 3 to 5 minutes. Stick the strawberry mixture in the refrigerator to chill.
When ready to serve, remove the sides of the springform pan and place the frozen cheesecake on a cake plate. Spoon the strawberries over the top.
For clean cuts, dip your knife in hot water and dry between slices.