Remove extra papery skin from the garlic head and trim off the top third to expose the tops of the cloves.
Drizzle the exposed garlic top with 1 tablespoon olive oil and wrap tightly in foil. Set aside.
Cut the corn kernels away from the cob into a shallow bowl.
Toss the corn with 1 tablespoon olive oil, smoked paprika, cumin, black pepper, and ¼ teaspoon of the sea salt.
Line a baking sheet with parchment paper and spread the corn in a single layer. Place the corn and the wrapped garlic in the oven.
Roast the corn for 15 minutes until slightly charred, and roast the garlic for an additional 15 minutes until very soft and golden brown.
Once the garlic is cool enough to handle, squeeze the cloves out of the skin.
Mash half of the cloves into a paste using a fork in a small bowl. Reserve the remaining cloves.
In a stand mixer bowl, stir the honey into 1 ⅔ cups warm water.
Sprinkle the yeast over top and let sit for 5 minutes until foamy.
Add 4 cups of bread flour, 2 tablespoons olive oil, 1 ¾ teaspoons sea salt, and the roasted garlic paste to the bowl.
Use the dough hook attachment to knead the dough until it forms a ball, adding the additional ½ cup flour as needed to keep the dough from being too sticky.
Remove the dough from the bowl and knead until smooth on a surface dusted with bread flour.
Oil the mixer bowl with 1 tablespoon olive oil and add the dough, turning to coat with the oil.
Cover the dough with plastic and rise until doubled, about 45 minutes to 1 hour.
Coat a baking sheet with 2 tablespoons olive oil.
Punch the dough down and transfer it to the pan, turning once to coat with the oil.
Gently stretch the dough outward until it forms an 11x14-inch rectangle. Allow the dough to relax for a few minutes in between stretching, if needed.
Cover the dough and let rise at room temperature until very puffy, about 20 minutes.
Preheat the oven to 425 F.
Drizzle the remaining 1 tablespoon olive oil over the proofed dough and scatter over the reserved roasted garlic cloves.
Use your fingers to dimple the dough, pressing firmly all the way to the bottom of the pan.
Spread the corn onto the dough, followed by the cotija cheese and thyme leaves.
Bake the focaccia for 25 to 30 minutes, until golden brown and the toppings are toasted.
Remove the focaccia from the oven and use a large spatula to transfer it to a cooling rack.
Cool the focaccia before cutting into squares and serving.