Slice section of baguette into desired sizes for two sandwiches, and then slice each section in half, lengthwise. Use ½ tablespoon to butter inside and outside the baguette. Place other ½ tablespoon butter in a cast iron skillet. Optional: Keep another cast iron skillet nearby to press your grilled cheese sandwich later on.
Melt ½ tablespoon butter in cast iron skillet over low flame. Spread 1 spoonful of tomato jam on the bottom half of each baguette section. Top with sliced cheddar. Add sliced cheddar to the other, buttered half. Place all four slices of baguette, cheese side up, on the cast iron skillet. Cover with lid and let cook about 8-11 minutes, checking occasionally to ensure nothing is getting too toasty.
At this point, cheese should be beginning to melt. If using second cast iron skillet as a press, begin to heat it on a second burner over low heat. While it warms, spread top halves of baguette with a spoonful of tomato jam each, and add a light handful of arugula to the bottom half of each sandwich. Place top half of baguette on bottom half. Remove second cast iron skillet from second burner, turn off both burners, and place second skillet down directly on top of the assembled baguette sandwiches. Let rest and cook about 1 minute, until cheese is melty and tops and bottoms of baguettes are perfectly toasty.
Remove from heat and eat immediately, preferably with some delicious Naked Pressed.