In a mixing bowl, whisk together the vanilla, rum, eggs, egg yolks, and 1/4 cup canola oil. In another bowl, sift the flour, baking powder, and sugar. Add the dry ingredients to the wet ingredients slowly and mix the whole time with an electric mixer. (I use my Kitchen-aid.) Beat until mixture forms a soft dough. Put plastic wrap over it and let it rest for 30 minutes.
Divide dough in half. Form each half into ropes 1/2 inch in diameter. Cut each rope into 3/4 inch pieces. Roll each piece into a ball.
In a pot or saucepan, heat 3 inches of oil. Add dough balls in batches and fry about 2 minutes until gold in color. Transfer to a platter lined with paper towels.
In another saucepan, combine the honey, sugar, and cinnamon stick. Cook the mixture until the sugar is dissolved, stirring often. Simmer for a couple of minutes. Remove the cinnamon stick and stir in the dough balls. Transfer the strufolli to a serving dish and serve at room temperature.