For the testaroli batter: In a large bowl, add the sparkling water and salt, and whisk in the flour until smooth and well blended. The batter should be about the consistency of light cream. Add more water if necessary to achieve the right consistency. Refrigerate and let rest for 30 minutes.
For the pesto: While the batter is resting, make the pesto. In a small pot or skillet, heat 1/4 cup of the olive oil over low heat and add the garlic. Heat to allow the garlic to infuse into the oil, about 3 minutes. Let cool to room temperature, and then discard the garlic.
In a food processor, add the pine nuts and pulse until coarsely ground. Add the basil, cheese and salt. With the processor running, drizzle in the garlic-infused olive oil and then the remaining 1/4 cup olive oil until completely combined. Transfer to a bowl and set aside.
Preheat the oven to 300 degrees F. Fit a baking sheet with a rack.
Heat a griddle or small nonstick skillet over medium heat and brush with some olive oil. With a ladle, pour in about 1/4 cup of the batter and gently swirl to form a circle, like a crepe. Cook until lightly browned, about 3 minutes. Flip and cook the other side until lightly browned and slightly crispy. Remove to the prepared baking sheet and put in the oven. Repeat with the remaining batter, making sure to brush the griddle with olive oil between each batch.
When done, remove the testaroli from the oven; they should be lightly crispy. If they are soft, place them on the griddle briefly to crisp up. Spread each testaroli with about 1 tablespoon of the pesto and serve immediately.