Combine the bread crumbs and the thyme.
Slice the goat cheese into 12 (½ inch thick) slices. The cheese tends to stick to the knife so using (unflavored :)) dental floss is a clean and easy way to cut it.
DIp each slice into the beaten egg and then into the bread crumbs making sure that each cheese slice is completely coated.
Place the slices on a rack and chill in the refrigerator for at least 15 minutes. (This can be done up to 4 hours ahead of time and kept in the refrigerator.)
When ready to serve, heat 1 Tablespoon of butter and 1 Tablespoon of olive oil in a saucepan over medium-high heat until just smoking.
Cook the goat cheese rounds on both sides until browned on the outside (less than 1 minute on each side).
Lay on top of the prepared salads.