Finely chop enough green garlic to make 1/4 cup. Cut more green garlic into 3-in.-long slivers to make 1/4 cup; set aside.
Stir together chopped green garlic, cheese, parsley, mint, and 1/4 tsp. each salt and pepper in a small bowl.
Heat 1 tbsp. oil in a small frying pan over medium-high heat. When hot, add slivered green garlic and sauté until about half of it begins to brown, 1 to 2 minutes. Transfer to paper towels to drain. Stir remaining oil from pan into cheese mixture.
Toast bread and spread with cheese mixture. Lay watercress on each, then drape prosciutto to fit. Top with green garlic slivers. Cut sandwiches in half. Drizzle remaining 1 tbsp. oil over sandwiches and sprinkle with more salt and pepper if you like.
Tips for cooks
Green garlic is simply immature regular garlic and tastes fresher and more delicate. It has juicy, edible stalks with no bulbs at the base (if picked quite young), or bulbs with tiny cloves that don't need peeling.
FIND Look for slender pale green stalks in springtime at farmers' markets and some supermarkets. Or substitute green onions plus regular garlic (see the recipe).
PREP Trim green garlic as you would green onions, removing tough tops (save to flavor soup) and roots, plus any tough outer layers. Ten stalks yield 2 cups chopped.
*If you can't find green garlic, use 4 green onions (half chopped and half slivered) plus 2 minced garlic cloves.