Heat the butter in a large skillet over medium-low heat. Add the onions and cook for 12 to 15 minutes, stirring occasionally, until soft and lightly golden.
Remove from the heat and let the onions cool slightly. Heat the oven to 375 F and grease a 10-inch skillet or baking dish.
Mix the Batter:
Place the cornmeal in a large bowl and stir in the warm milk. Rest for 10 minutes so the cornmeal hydrates.
Whisk in the eggs, salt, and pepper. Fold in the softened onions and about two-thirds of the cheese.
Bake and Serve:
Pour the batter into the prepared skillet or baking dish and smooth the top. Sprinkle with the remaining cheese.
Bake for 40 to 45 minutes, until the top is golden and the center is set. Cool for 10 minutes before cutting into squares.
Notes
Medium-grind cornmeal gives the bread a pleasantly rustic crumb.
Do not rush the onions; their sweetness balances the savory cheese.
A mild, well-melting cheese is the best substitute for queso Paraguay.