Heat water and canola oil until warm (120°–130°F). Add to flour mixture. Add vinegar. Blend at low speed until well combined.
Switch to the dough hook and mix in the remaining flour a little at a time, to make a soft dough that’s not sticky to the touch, adding more or less flour as needed.
Knead the dough for 5 minutes. Place in a greased bowl, turning to grease the top. Cover; let rise in warm place until almost double.
Preheat oven to 400º F. Grease a 9-inch springform pan, and line bottom with parchment; grease paper. Place on top of a baking sheet. Set aside.
Punch down the dough. On a floured surface, roll the dough into very thin rectangle, as thin as you can. Spread a thin layer of pesto on top of the dough. Start at the top of the long edge and slowly, tightly and gently roll the dough into a log. Pinch it closed.
Braid bread according to recipe directions. Carefully pick up the braid and place it in the prepared springform. Cover; let rise in a warm place until almost double.
Bake at 400º for 5-10 minutes, lower oven temperature to 350º and bake for an additional 20-30 minutes. When the bread is out of the oven lightly brush olive oil on top and sides. Let cool on a rack.