Pulse shrimp and all ingredients, except the phyllo and butter, in the bowl of a food processor until it is a smooth paste. You may need to do this in small batches.
Transfer shrimp paste to a small bowl that is set inside a larger bowl filled with ice. It is very important to keep the shrimp mixture very cold to eliminate any food safety concerns.
Place phyllo dough on a sheet tray and cover it with a lightly dampened, clean kitchen towel or paper towel. You may want to prepare these in small batches right through the frying process to keep the filled phyllo dough from becoming soggy and possibly sticking. If they do stick a bit or break before they are fried, have no fear. They will cook up the same and still be delicious. It will only give them a more rustic look, which is fine. Remember to always refrigerate any mixture not being used until you are ready for it.
Pipe a line of filling with a 1/2" tip or spread about 1 1/2 to 2 tablespoons of shrimp mixture across the bottom of the phyllo dough, allowing for about a half inch border on the bottom and sides.
Lightly brush the top 1/2" with melted butter and roll to close. Make sure it is sealed well on the top and use an extra dab of melted butter if needed.
Set on a sheet tray lined with parchment or wax paper and cover with a lightly dampened towel.
Pour canola oil into a large dutch oven or any other heavy bottomed pot. Attach a thermometer to the edge and make sure the bottom of it does not touch the bottom of the pot for an accurate reading.
Heat the oil on medium-high to high heat to 365 degrees F. Carefully place a few shrimp logs into the oil at a time, using a slotted spoon if necessary. Watch carefully, turning once or twice while frying to make sure that all sides are golden brown, as they ook quickly.
Remove with spider or slotted spoon and transfer to a cooling rack that has been placed inside a paper towel lined sheet pan.
Repeat process until all shrimp paste is used and logs have been fried.
They are best when served right away, but if not, they can be kept warm and crispy in a 200 degrees F oven.