Preheat your broiler.
Toss the eggplant with the salt. Place in a colander over a bowl to catch any liquid run-off. Tumble occasionally. Let the eggplant sit for 10 minutes or so before using.
Heat a large cast-iron or other heavy-bottomed skillet over medium-high heat. Add the olive oil and swirl.
When the oil wrinkles into a semi-psychedelic haze, just before it reaches the smoke point, add the eggplant in a single layer. Let cook for 2 minutes before attempting to stir.
Now grab a wooden spoon. Sauté the eggplant for about 5 minutes or until well browned and tender, stirring occasionally. Add the yellow onion and the garlic. Stir.
Lower the heat to medium-low. Add the sugar, sherry vinegar, and thyme. Fold to integrate, and cook over low heat for 10 minutes, until the mixture softens and becomes unified.
Turn off the heat. Let the eggplant mixture rest for a few minutes--right there in the pan.
Place the bread slices on a baking sheet and broil until slightly charred on both sides, 1 minute per side. Remove the bread. Put the walnuts on the baking sheet in a single layer.
Broil the walnuts until the first sign of char appears on a single nut piece. Remove from the oven and fold into the eggplant mixture. Add the parsley to the eggplant mixture.
To serve, spoon a generous amount of eggplant mixture on the bread slices. Sprinkle evenly with the pecorino Romano.