Melt the fats: In a large skillet over medium heat, melt the butter along with the olive oil. Stir in the diced onions and water, covering the skillet for about 5 minutes to soften the onions.

Caramelize the onions: Reduce the heat to medium-low, and stir the onions frequently for about 1 hour, until they're deeply caramelized and golden brown. Transfer to a bowl, and stir in fresh thyme.

Combine the dough ingredients: In a large bowl, mix together the flour, lukewarm water, parmesan cheese, olive oil, sugar, yeast, and salt using a wooden spoon until completely combined.

Let the dough rise: Lightly oil your hands and a separate large bowl. Place the dough into this bowl, cover it with plastic wrap, and let it rise in a warm place until it has doubled in size, about 1 hour.

Prepare the garlic oil: While the dough is rising, heat olive oil in a skillet, then add the crushed garlic. Cook for 5 minutes until fragrant and lightly browned, then remove and discard the garlic.

Preheat the oven: Set your oven to preheat at 425°F (220°C). Pour half of the garlic oil into an unlined metal 9×9 baking pan and set it aside.

Shape the dough: Once the dough has risen, lightly oil your hands. Stretch and fold each corner of the dough toward the center to create a round shape. Place seam side down in the oiled pan, cover it, and let it rise again for about 1 hour.

Prepare for baking: After the second rise, the focaccia should be jiggly and bubbly. Pour the remaining garlic oil over the top, then use your fingers to press deep dimples into the dough. Sprinkle with remaining thyme.

Bake the focaccia: Transfer the pan to the oven and bake for 12-18 minutes, until the top is golden brown and a thermometer reads at least 190°F (88°C).

Finish and enjoy: Once baked, spoon the caramelized onions over the top, let it cool on a wire rack, and garnish with flaky sea salt and extra thyme. Cut into servings and enjoy warm!