Heat a large skillet on medium heat.
If your skillet is large enough to lay out 4 slices of bread, add half of the butter to the skillet. If your skillet is just large enough for 2 slices of bread at a time, use about a quarter of the butter (a little less than a tablespoon).
As soon as the butter melts, lay the pieces of bread on it and move them around a little so they soak up the butter.
Let cook until the bottom sides of the bread are toasted.
Once the bottom sides of the bread are toasted, turn them over, slide one of the pieces over to the far side of the pan and stack the remaining piece on top of it for just a moment while adding more butter.Add half of the remaining butter to the pan. Put one of the slices of bread over the butter, toasted side up.Then stack the other slice of bread on top of the piece that is lying on the melted butter.
Add the remaining butter to the pan, and as soon as it melts lay the remaining piece of bread over it toasted side up.
To the top of each slice of bread, add cheese.
When the bottom sides of the bread are lightly browned, add ½ cup of warm chili over the cheese on one slice of bread (add 1 cup of chili on two slices if cooking all four pieces of bread at the same time).
Add 2 more slices of cheese on top of the chili.
Then make the sandwiches out of the slices of bread and cheese (top the chili with the non-chili slices of toasted bread).
Cover the pan with a lid and turn the heat off while the cheese melts.
When the cheese is completely melted, transfer the sandwich onto a cutting board, cut it in half and serve right away while warm.