In a food processor, pulse together the flour and sugar to combine. Add the butter pieces, and pulse until the mixture looks like coarse meal with some pea-size pieces.
Sprinkle with 4 Tbs. of the water, and pulse until the dough comes together. If too dry, add the remaining water by the teaspoon, and knead until the dough comes together and is soft, smooth, and not crumbly.
Form the dough into a large ball and wrap the dough in cling wrap. Press down into a thick disk and set in the fridge to chill for 60 minutes.
For the Galette Assembly
20 minutes before removing the dough from the oven, preheat the oven to 400º and line a baking sheet with parchment paper.
Take the dough out from the fridge and lay on a flour-dusted surface. Roll out the dough until approximately 11” in diameter and approximately ¼” thick.
Slice the heirloom tomatoes into thin slices, then set down on a paper towel to pat dry.
To the dough, place the sliced tomatoes into the center of the dough overlapping on each other as needed.
Fold the outsides of the dough on top of the galette, overlapping each fold.
In a small bowl, crack the egg and whisk. Brush the crust of the galette with the egg wash.
Bake for 30 minutes. Let cool for 15 minutes before slicing and serving.