1. Put a rimmed 10- by 15-inch pan in the freezer until cold, at least 10 minutes. Shape ice cream into 18 egg-shaped ovals: to make each portion, dip an oval ice cream scoop in water, drain, and quickly scoop ice cream into a 3-tablespoon-size oval. As shaped, put each oval on cold pan. (Or use a round scoop, shape ice cream into balls, and set on pan. With a soup spoon, quickly press balls into ovals.) Immediately return pan to freezer. If storing longer than 2 hours, wrap airtight when ice cream is hard and freeze up to 2 days.
2. Cut 6 sheets of foil, each 12 by 18 inches. From a narrow side, fold each sheet in half.
3. As a form, use a bowl about 2 1/2 inches deep and 5 to 5 1/2 inches wide. Center 1 foil rectangle over bowl, then press smoothly into bowl and fold foil ends over bowl rim. Lift out foil bowl and repeat to shape remaining foil rectangles. Set foil bowls on 2 baking sheets, each 12 by 15 inches.
4. Lay 1 filo sheet flat. Keep remaining filo dough covered with plastic wrap to prevent drying. With a pastry brush, quickly and lightly streak filo sheet with butter. Sprinkle evenly with 1 tablespoon sugar. Top with another filo sheet, brush with butter, sprinkle with sugar, and repeat to make 1 stack of 4 filo sheets.
5. From a long edge, fold filo stack in half. With a long, sharp knife, cut stack across the narrow width of filo to make 1/8-inch-wide strands. Divide strands into thirds.
6. Use a third of the cut filo to make 1 nest: Gently place a handful of cut filo into bottom of 1 foil bowl. Then gently drop remaining filo, bit by bit, against bowl sides, spiraling up to rim. Leave a few strands sticking up randomly around rim. Repeat to make 2 more nests.
7. Repeat steps 4 through 6 to shape 3 more nests.
8. Bake nests in a 350° oven until golden brown, about 15 minutes (10 minutes in a convection oven); switch pan positions halfway through baking.
9. Let nests cool in foil bowls on a rack. With a small metal spatula, gently ease nests from foil.
10. Set nests on dessert plates or in bowls. Place 3 ice cream ovals in each nest. Add apricot sauce to taste.