Trim and discard ends of green onions. Chop onions and mix with rice, cheese, paprika, caraway, and capers.
Lay 1 filo sheet flat (cover remaining filo with plastic wrap to prevent drying) and brush lightly with butter. Top with another filo sheet and brush lightly with more butter. Repeat to stack remaining filo.
Starting about 1 inch from a narrow edge, lay ham slices over filo (overlapping as needed) to cover about 1/3 of dough, leaving a 1- to 2-inch margin on the sides. Evenly pat rice mixture over ham. Roll filo from ham end to enclose filling. With seam down, gently transfer strudel roll to a buttered baking sheet (at least 11 by 14 in., without side rims). Lightly brush strudel with butter.
Bake in a 400° oven until golden brown all over, 15 to 17 minutes.
Meanwhile, mix sour cream with mustard in a small bowl. Mix watercress with vinegar.
Using 2 wide spatulas, slide strudel onto a platter, cut into thick slices, and accompany with watercress salad. Add sour cream sauce to taste.