Rinse, hull, and roughly chop the strawberries. Toss with 1 teaspoon sugar.
Soften cream cheese briefly in the microwave, then stir in powdered sugar until combined.
Flatten bread slices with a rolling pin and trim off the crusts.
Spread about 1 teaspoon cream cheese mixture along one edge of each bread slice. Top with about a spoonful of strawberries, then roll tightly.
Whisk eggs with milk, remaining 2 teaspoons sugar, and cinnamon in a shallow bowl.
Heat a nonstick pan over medium heat with a little oil and butter.
Coat each roll in the egg mixture and place seam-side down in the pan. Turn a quarter turn as each side browns until cooked and evenly golden.
Dust with powdered sugar if desired and serve with remaining strawberries.