Triple-Threat Onion Galette

Whisk sugar, salt, and 1½ cups flour in a medium bowl to combine. Add butter and toss to combine. Using your hands and working quickly, rub and smash butter into flat, irregular pieces (be careful not to overwork; you don’t want to soften the butter too much).1 Drizzle ⅓ cup cold water over and mix with a rubber spatula, smashing in butter, until dough mostly comes together—it will be dry and shaggy.
Turn out dough onto a lightly floured surface. Working quickly and using a rolling pin and your hands, press dough together to form a mass. Roll dough into a rough square about ½" thick. Fold in half into a rectangle, then once again to make a small square. Press down on the dough with rolling pin to make it hold together. Roll dough out once more to make a rough square about ½" thick. Repeat folding process, taking your time to make it neat as this will be your final fold. Using rolling pin, gently flatten dough, rotating as needed, to make a 6"-diameter disk.
Dust surface with more flour and roll disk into a 14"-diameter circle or oval about ⅛" thick. Fold dough in half, then in half again to make a rough triangle (this will make it easier to move). Slide onto a parchment-lined baking sheet and unfold. Chill while you make the filling.
Do Ahead: Dough can be made 3 days ahead. Wrap 6" disk tightly and chill until ready to roll out.
Place a rack in middle of oven; preheat to 375°F. Heat 3 Tbsp. butter in a medium skillet over medium until foaming, about 1 minute. Reduce heat to medium-low; add onion, garlic, and 1 tsp. Diamond Crystal or ½ tsp. Morton salt, and stir to combine. Cover and cook, stirring occasionally, until onion is tender and just beginning to brown, 6–8 minutes. Uncover and continue to cook, stirring occasionally and adding a splash of water if onion begins to stick, until golden brown in spots, about 5 minutes more. Transfer onion mixture to a plate; reserve skillet.
Melt remaining 1 Tbsp. butter in reserved skillet (no need to wipe out) over medium heat. Remove from heat and add scallions and remaining ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt. Toss scallions until coated in butter.
Remove dough from refrigerator and, using a small offset spatula or a spoon, spread mustard in a thin layer over dough, leaving a 1½"–2" border around the edges. Scatter onion mixture over mustard and sprinkle with Parmesan; season with pepper. Cutting from edge of dough to the start of the mustard and filling, make slits in border of galette at 2"–3" intervals. Fold flaps up and over filling, slightly overlapping. Scatter scallions over. Bake galette until crust is deep golden brown and scallions are lightly charred, 45–50 minutes. Using parchment paper, slide galette onto a cutting board and cut into wedges.
Do Ahead: Onion can be cooked 3 days ahead; transfer to an airtight container, then cover and chill. Galette can be assembled 1 hour ahead; chill.