Combine strawberries and sugar in a blender and purée until smooth.
Strain the mixture through a fine-mesh sieve into a large saucepan.
Bring the purée to a boil, then reduce it to a simmer and cook over medium-low heat, stirring occasionally for 45 minutes to 1 hour. Stir it more frequently at the end until the mixture has reduced to 1 to 1¼ cups. It should be thick enough to fold over itself.
Preheat the oven to 200°F and line a large baking sheet with nonstick liner, such as a Silpat.
Pour the hot purée onto the liner and use an off-set spatula to spread it thinly (as evenly as possible) into a 15"x10" rectangle.
Dry the purée in the oven until it feels slightly tacky but does not stick to your fingers, roughly 2 to 3 hours.
Transfer the roll-ups (still on the Silpat) onto a cooling rack to completely dry, at least 3 hours (and up to 24).
Place a piece of parchment paper on top of the fruit roll-ups and then peel off the paper and the roll-ups together. Roll them up in the parchment, or cut with cookie cutters into desired shapes.