Whisk together the warm water, yeast, sugar, and melted butter in a small bowl; let stand until yeast is dissolved, about 10 minutes.
Whisk 1 1/4 cups flour and salt together in a medium bowl. Add yeast mixture to flour mixture and stir with a wooden spoon until mixture clumps together. Mix until incorporated. Add more flour, a little at a time, if dough is too wet.
Turn dough out onto a lightly floured work surface and knead until smooth and elastic, about 3 minutes. Divide dough into 12 equal pieces; roll each into a ball. Place on a flour-dusted baking sheet; cover with plastic wrap. Let rise in a warm place until doubled in size, about 45 to 60 minutes. (Dough can be made up to this point 1 day ahead. Keep covered and refrigerate.)
For caramelized onions:
Heat oil in a large nonstick skillet over medium-high heat. Add onion slices to skillet and season with 1/2 teaspoon kosher salt. Cook onions without turning until they are softened and beginning to brown, about 5 to 7 minutes. Add sherry vinegar and honey and stir well to incorporate; reduce heat to medium and continue to cook until onions are tender and brown, occasionally stirring gently, about 10 minutes. Season, to taste, with salt.
Remove skillet from heat and cool. (Onions can be prepared up to 1 day ahead. Cover and refrigerate.)
Preheat oven to 450 degrees F.
Sprinkle 2 baking sheets lightly with cornmeal. Roll out each dough ball to 3 to 4-inch rounds. Transfer 6 rounds to each prepared baking sheet. Pierce dough all over with fork. Brush generously with extra-virgin olive oil. Arrange onions evenly atop each round, leaving 1/4-inch plain border. Top with thyme leaves. Bake flatbreads until crusts are puffed and golden, about 12 to 15 minutes. Sprinkle crumbled goat cheese on top of each mini focaccia. Transfer to serving platter and serve immediately.