Chop basil and dill together until finely diced. Place in a medium bowl. Add goat cheese, honey, lemon juice, egg, and salt together and whisk together until combined. Cover and refrigerate for at least 30 minutes to firm up the cheese again.
Once cheese is firmed, preheat oven to 375 degrees. Prepare baking sheet by lining with parchment paper. Set aside.
Remove phyllo sheets from refrigerator. Remove one sheet to work with and cover the stack of the rest with a sheet of plastic wrap or damp towel. Generously brush the entire sheet with olive oil. Place another sheet of phyllo on top to create a second layer. Brush with more olive oil. Top with another sheet of phyllo dough. Slice the phyllo dough down the center of the sheet. Cut perpendicularly down the center of the other edge. You should have 4 even rectangles.
Place 1 tablespoon of goat cheese mixture in the bottom short edge of the rectangle. Gently roll the dough into a cigar, starting at the edge with the filling. Do not roll too tightly. Repeat with the other pieces of phyllo. Make sure cigars are separated at least 1 inch apart on the baking sheet. Brush each cigar with more olive oil
Bake cigars in preheated oven for 15-18 minutes, or until the tops are golden brown. Some filling may escape but that's to be expected.
Remove baked cigars from oven and let them cool on the baking sheet for 10 minutes before serving.