Preheat the oven to 400°. Heat the oil in a large skillet. Add the spinach and garlic in batches and cook over moderate heat until the spinach wilts. Transfer to a colander and squeeze out all the water; if the spinach isn't dry, the rolls won't be crisp. Chop the spinach and mix it with the sun-dried tomatoes, olives and cayenne.
Lay a sheet of phyllo on a work surface with the short side directly in front of you. Brush with butter and sprinkle with 1 tablespoon of the Parmesan and a pinch of nutmeg. Spread about 1/4 cup of the spinach filling in a narrow strip across the short edge of the phyllo. Lay 4 anchovy halves over the filling and roll up tightly. Brush the roll with butter and sprinkle with 1 tablespoon of the Parmesan. Place on a baking sheet. Repeat with the remaining ingredients, leaving ample space between the rolls. Bake on the top shelf of the oven for 10 to 12 minutes, or until golden brown and crisp. Let the rolls cool slightly before slicing at an angle into bite-size pieces.