Heat broiler. Place apricot halves, cut side up, on one half of a baking sheet. Brush tops with 2 tablespoons of butter; sprinkle with 3 tablespoons of sugar.
In a small bowl, toss pecans with remaining butter and sugar and spread them out on the other half of the baking sheet. Broil for 2 to 3 minutes, watching carefully, until apricots begin to brown and pecans are lightly toasted.
Place three apricot halves on each plate, drizzle with warm dulce de leche, and sprinkle with the pecans.