Heat the oven to 425°F and line a sheet pan with parchment. Brown the beef and onion in a skillet over medium-high heat for 5 to 6 minutes, breaking up the meat and bringing it to 160°F.
Stir in tomato paste, paprika, cumin, salt, olives and water. Cook for 1 minute until nearly dry, then spread on a plate to cool for 3 minutes.
Divide the filling among the pastry discs, leaving a 1/2-inch border. Moisten the edges with a little beaten egg, fold into half-moons and crimp firmly with a fork.
Arrange on the sheet pan, brush the tops with the remaining egg and cut one small vent in each. Bake for 12 to 14 minutes, until golden and the filling reaches 165°F.
Toss the salad greens with olive oil and vinegar. Serve two empanadas per person with the salad.