Bring water to a boil in a medium saucepan. Remove from the heat, add garlic, stir to submerge and let sit until the garlic skins are softened and cool enough to handle, about 50 minutes. Strain the garlic, remove the skins and cut off the hard nub where the clove was attached to the head.
Place the garlic, canola oil and extra-virgin olive oil in a medium saucepan; bring to a gentle simmer over medium-low heat. Reduce the heat to low and maintain a very gentle simmer (it may be necessary to slide the pan to the edge of the burner). Simmer until the cloves are golden and very soft when pressed with a fork, 40 to 50 minutes. Let cool for 30 minutes.
Transfer the cooled garlic to a sieve to drain, reserving the oil. Transfer the garlic to a food processor and puree until smooth, scraping down the sides occasionally. Store the puree and the oil separately in the refrigerator.