Pulse together all filling ingredients in a food processor until they form a coarse paste.
Preheat oven to 450°F.
Cover phyllo with 2 overlapping sheets of plastic wrap and then a dampened kitchen towel. Put 1 phyllo sheet on a work surface, keeping remainder covered, and brush generously with butter. Stack 2 more phyllo sheets on top, generously buttering each.
Cut phyllo stack in half lengthwise, then arrange 1 half with a long side nearest you. With dampened fingers shape 3 tablespoons shrimp filling into a narrow rope along edge nearest you, then roll up paste tightly in phyllo to form a long thin roll. Make another roll with other half of stack. Make 6 more rolls with remaining phyllo sheets, butter, and shrimp filling.
Lightly brush top of rolls with egg and sprinkle with sesame seeds and kosher salt. Cut each roll crosswise into 4 sticks and arrange, seam sides down, 1 inch apart on 2 large baking sheets. Bake sticks in upper and lower thirds of oven, switching position of sheets halfway through baking, until phyllo is golden brown, about 15 minutes total.
Transfer sticks to a rack to cool slightly and serve warm with dipping sauce.