Preheat the oven to 350 degrees F.
Brush a sheet pan with melted butter.
In a food processor, pulse the walnuts and pistachios until coarsely chopped. Transfer the nuts to a large bowl. Put the apricots in the processor and pulse them until coarsely chopped, add to the nut mixture and mix well with your hands to combine. Add 2 tablespoons of the honey and 1/4 cup of the melted butter; toss.
Unroll the phyllo dough and lay the sheets flat on a work surface. Keep the pastry covered with a damp, not wet, towel as you work to prevent drying out. Place 1 sheet of phyllo on the buttered pan and brush it with more melted butter and sprinkle with about 1 teaspoon of sugar. Repeat with the 3 remaining sheets of phyllo, stacking each on top of each other and sprinkling each with sugar.
Spread the nut filling in a thin layer on top of the pastry, leaving a 1-inch border on all sides. Fold in the long sides and roll up the pastry like a cigar. Brush the entire top with even more melted butter and sprinkle the pastry log with the remaining teaspoon of sugar. Bake for 45 minutes, or until golden brown and crispy.
Slice the borek into 1-inch pinwheels. Drizzle with the remaining 2 tablespoons of honey before serving.