Special equipment: a silicone baking mat
Combine the egg whites and confectioners' sugar in a small bowl. Mix with a fork to avoid incorporating air, then slowly add the melted butter, then the honey. Add the flour and stir until smooth. Place the bowl in the refrigerator for 1 hour, then use right away. (If storing for later use, make sure you let the batter come to room temperature for 1 hour before use.)
Preheat the oven to 325 degrees F and line a half-sheet baking pan with a silicone baking mat.
Spread 2 tablespoons of batter on the silicone liner using the back of a spatula to make a perfect 5-inch circle. Repeat to make a second circle.
Bake the ice cream cones until golden brown, about 10 minutes. Using an offset spatula, immediately but gently lift the batter off the pan and roll into a cone, overlapping the edges and pressing the edges firmly to seal. (If you need to adjust the cones after they have cooled, place them back in the oven for 20 seconds and they will become soft and pliable again. Transfer to a rack to cool completely, about 30 minutes.
Repeat with the remaining batter.