To make phyllo tartlet shells: Preheat oven to 325°F. Lightly coat 2 mini-muffin pans with cooking spray.
Whisk together egg white, oil and salt in a small bowl.
Lay a sheet of phyllo on a work surface and with a pastry brush, lightly coat it with the egg-white mixture. Lay a second sheet smoothly on top, taking care to line up the edges before setting the sheet down. (Once you set down the sheet, it cannot be moved.) Brush with the egg-white mixture and repeat with 1 more sheet. Lay a fourth sheet on top but do not brush with the egg-white mixture.
Cut the dough into 4 strips lengthwise and 6 strips crosswise with a knife, making 24 squares. Press squares into the prepared muffin cups and bake until golden brown and crisp, 8 to 12 minutes. Transfer the tartlets to a rack and let cool. Repeat the procedure with the remaining 4 sheets of phyllo and egg-white mixture.
To make smoked trout filling: Combine cream cheese and smoked trout in a food processor; process until fairly smooth. Add scallions and horseradish and pulse until just combined. (Alternatively, finely mince the smoked trout with a knife and combine with the cream cheese, scallion and horseradish in a small bowl.)
Shortly before serving, spoon or pipe about 1 heaping teaspoon of filling into each tartlet shell and garnish with shredded cucumber.
Make Ahead Tip: Store the baked tartlet shells (Steps 1-4) in an airtight container at room temperature for up to 1 week or in the freezer for up to 2 months. Cover and refrigerate the filling (Step 5) for up to 2 days.