Place the cottage cheese in a fine-mesh strainer set over a bowl and drain for 30 minutes. Press firmly with a spoon to squeeze out as much liquid as possible.This step is the secret to syrniki that hold their shape and turn creamy inside instead of wet.
Pulse the drained cottage cheese in a food processor for a few seconds until fine and mostly smooth. Transfer to a bowl and stir in the egg, sugar, vanilla extract and salt, then mix in 1/2 cup flour until a soft, slightly sticky dough forms.
Spread the remaining flour on a plate. Divide the dough into 10 portions, roll each in flour and gently flatten into a 1/2-inch thick round, smoothing the edges with your fingers.
Heat the butter and oil in a large nonstick skillet over medium-low heat. Cook the syrniki for 3 to 4 minutes per side until deep golden brown.
Reduce the heat to low, cover the pan and cook for 2 minutes more so the centers cook through and turn custardy.
Serve warm with a spoonful of Greek yogurt, fresh berries and a drizzle of honey.