In a small bowl, sprinkle the gelatin into the stock; let sit until gelatin softens. In a large saucepan, heat the EVOO over medium-high. Add the onion, season with salt and pepper and cook, stirring often, until golden brown, about 10 minutes. Reduce the heat to medium; add the thyme and bay leaf. Cook, stirring often, until the onion is deep golden brown, about 20 minutes longer.
Add the stock mixture. Bring to a simmer, stirring and scraping the bottom of the pan, until the gelatin dissolves, about 5 minutes. Season. Pour into an 8-inch baking dish. Refrigerate until set, 2 hours.
Preheat the broiler. Coat a 24-cup mini-muffin pan with cooking spray. Working in batches, brush each gyoza wrapper with water. Scoop 1 1/2 tsp. gelatin mixture onto the center of the wrapper. Fold in half, pressing gently around the filling to seal the edges. Nestle the dumplings, seam side up, into the muffin cups. Repeat with the remaining wrappers and gelatin mixture.
Sprinkle the cheese over the dumplings. Broil until the cheese melts and starts to brown, about 5 minutes. Remove from the oven, top with the bread cubes and broil until the bread is toasted, 2 to 3 minutes.