In a large bowl, whisk the flour, sugar and salt. With a pastry cutter or your fingertips, incorporate the butter into the flour mixture until it resembles coarse meal, with some larger chunks here and there.
Form a well in the center, and add in 4 tablespoons of ice water. Mix with your fingers until the dough holds together enough to form a ball. If it is too dry, add the rest of the water one teaspoon at a time.
Form a well in the center, and add in 4 tablespoons of ice water. Mix with your fingers until the dough holds together enough to form a ball. If it is too dry, add the rest of the water one teaspoon at a time.
In the meantime, turn on the oven to 400°F (204 C) and then start heating up the oil in a pan over medium heat. Add in the spinach, garlic and salt and pepper and cook until tender about 10 minutes. Transfer to a plate to cool and set aside.
Now start rolling the dough between two sheets of lightly floured parchment paper into a 14-inch (35.5 cm) circle. Make sure to loosen the parchment paper from the dough occasionally and add some extra flour if it sticks. Discard the top parchment paper and transfer the bottom parchment paper with the dough onto a baking sheet.
Leaving 2-inches (6 cm) at the border, spread the Dijon mustard over the dough, then scatter the cooked spinach and halved cherry tomatoes and sprinkle with the feta cheese.
Fold the edges of the dough over the filling. If you have any issue with the dough sticking to the parchment paper, refrigerate it for 5-10 minutes, and then continue folding it. Brush the folded dough with the egg wash and sprinkle with the shredded mozzarella and cheddar cheese.
Bake the galette for 35 to 45 minutes, rotating halfway through to ensure even color all around. Remove from the oven and let it cool for about 10 minutes, then cut and serve with a side salad or enjoy it on its own!