Preheat oven to 350 and line a baking sheet with parchment paper. Cut butter into small pieces, then put it back in the refrigerator.
In a dry skillet (not nonstick), over medium-high heat, saute the onions, stirring frequently. Do not allow the onions to burn. As the onions cook, a brown crust will develop on the bottom of the skillet. Add about 1/4 cup water and 1/2 tsp soy sauce to the skillet to deglaze the bottom, continuing to stir, until the water has cooked off. Repeat about 2 more times (without the soy sauce, unless you really want a lot of salty umami flavor), until the onions are soft and well browned. Set aside to cool.
In a medium bowl, whisk together flour, baking powder, and salt until combined. Remove the cut-up butter from the fridge and incorporate into the flour mixture as quickly as possible with a pastry cutter, two knives, or your fingertips, until the mixture resembles a coarse cornmeal. No pieces of butter larger than a pea should remain.
Shred 3 ounces of the gruyere cheese and thinly slice the other 3 ounces into 6 slices. In a small bowl, combine the shredded gruyere, greek yogurt, egg, and caramelized onions, stirring very well. Add yogurt mixture to flour, stirring gently, until it combines into a soft dough. You may need to knead a few times, but be gentle. You don't want to overwork the dough.
Either pat the dough into one large round, or divide in half and make two smaller rounds. Pat the rounds to a thickness of about 1 inch. If you are only making one round, cut into 8 equal pieces. If you are making two rounds, cut each into 6 equal pieces. Cut the 6 slices of gruyere on the diagonal into 12 triangles and lay one triangle across the top of each scone piece.
Bake at 350 for 20 minutes, or until the scones are firm and well browned and the cheese is melted and lightly browned. (Keep an eye on the cheese; you may need to briefly toast it under the broiler, but you definitely don't want the tops to burn.) Serve warm. The scones will keep, at room temperature, for four days.