For filling, in a large skillet melt 1/4 cup butter over medium-high heat. Add apples, walnuts, and dried cherries. Cook about 4 minutes or until apples are softened, stirring occasionally; remove from heat. In a small bowl stir together sugar and cinnamon. Sprinkle cinnamon-sugar mixture over apple mixture and toss gently to combine; cool.
Preheat oven to 375 degrees F. Line two large baking sheets with parchment paper; set aside. Unroll phyllo dough; cover with plastic wrap. (As you work, keep the phyllo covered to prevent it from drying out; remove sheets as you need them.) Place one sheet of phyllo on work surface; brush lightly with some of the 1-1/4 cups melted butter. Top with second sheet of phyllo. Repeat brushing and layering two more times (four sheets total). Brush top with melted butter.
Cut phyllo stack lengthwise into three strips. Place one cheese slice about 2 inches from an end of each strip, cutting cheese to fit if necessary; top with 1 rounded tablespoon of the filling. To fold into a triangle, bring a corner over filling so the short edge lines up with the side edge. Continue folding in this way along the length of the strip until the end is reached. Place triangles on one of the prepared baking sheets. Brush tops lightly with melted butter. Repeat with the remaining phyllo, butter, cheese, and filling.
Bake for 15 to 18 minutes or until crisp and golden; cool slightly. Transfer all the baked triangles to a 15x10x1-inch baking pan, overlapping slightly if necessary. Drizzle triangles with half of the Lemon-Honey Syrup. Let stand for 15 minutes. Drizzle with the remaining syrup. Let stand for at least 30 minutes more before serving.
In a medium saucepan, stir together sugar, the water, honey, lemon peel, lemon juice, stick cinnamon, and, if desired, rosemary. Bring to boiling; reduce heat. Simmer, uncovered, for 20 minutes. Remove cinnamon and, if using, rosemary.