In a large heavy-bottomed skillet over medium heat, add the oil and butter. When the butter is melted, add the onions, sugar, and salt, to taste. Cook until the onions begin to brown. Adjust the heat to low and cook slowly, turning often, until the onions are golden brown and caramelized, about 10 to 15 minutes. Remove the onions from the heat and set aside to cool.
Preheat the oven to 375 degrees F.
In a bowl combine the goat cheese, egg, cream, and thyme. Pour the mixture into a small resealable plastic food storage bag. Snip off a corner of the bag to use it for piping.
Lay the pie crusts out flat on a clean surface and with a 2 1/2-inch round cutter cut out 12 circles from each crust. Press them into the cups of the mini-muffin tin. Fill each cup 1/2 way with the cooled onions. Pipe the cheese mixture on top of the onions, filling the cups. Bake the tarts until the filling is set, about 15 to 20 minutes. Let the tarts cool slightly, and then remove them from the muffin tin and serve.