Prepare the pastry: place the flours, cane sugar, salt, thyme, pepper and butter in the food processor and pulse until the butter is in pea-like pieces, about 10-12 pulses.
Then, drizzle the water into the food processor while pulsing, just until dough comes together - it’s quick!
Flour your surface, and knead the dough a few times until it comes together in a flattened disk. If dough feels a bit too sticky, just sprinkle a little more flour. Wrap disk in plastic wrap and chill in the fridge for one hour.
Preheat the oven to 400℉ (205℃). Cut a large piece of parchment and lightly flour it. Place chilled dough on floured parchment and roll out until it's ¼ inch (½ cm) thick. Transfer the parchment to a rimmed baking sheet and place in the fridge for 15 more minutes while you prep the tomatoes.
Spread the chilled dough with pesto, leaving a 2 inch border around the edges. Then spinkle over the garlic and the sliced tomatoes.
Fold the border over the tomatoes, overlapping and gently pressing to seal the edges. If they won't stick, wet the dough with your fingertips and that should help.
Bake for 50-65 minutes, checking every 3 minutes after 50. You want the tomatoes looking bubly and the edges of the pastry golden brown (but not dark!)
Let it cool for 5 minutes then transfer to a cutting board and slice into squares, using a sharp knife or pizza cutter. Serve warm or room temperature.