Preheat the oven to 350°F (177°C) and line a sheet pan with parchment paper.
In a medium bowl, whisk together 2 cups flour, 3 tsp baking powder, 1 tbsp sugar, and 1/2 tsp salt until evenly combined.
Add 1 3/4 cups Greek yogurt to the dry ingredients and stir with a wooden spoon until the mixture comes together in clumps; do not overmix.
Turn the dough onto a lightly floured surface and gently gather it into a ball, adding 1–2 tablespoons more flour only if it is excessively sticky; press into a thick disk.
Cut the disk into 8 equal pieces and roll each piece into a smooth ball using your hands; place the rolls on the prepared pan spaced slightly apart.
Beat the egg in a small bowl and brush the tops of the rolls with the beaten egg.
Bake for 24–28 minutes, checking at 20 minutes and then every couple of minutes until the rolls are golden brown and cooked through.
While the rolls rest for a few minutes, melt 2 tablespoons butter and stir in 1 minced garlic clove and 1 tablespoon chopped parsley to make garlic butter.
Brush the tops of the warm rolls with the garlic butter and sprinkle with flaky sea salt and additional parsley if desired.