Preheat the oven to 375ºF.
In a large bowl, whisk together the flour, baking soda, salt and caraway seeds. Stir in the buttermilk and melted butter with a rubber spatula, scraping down the sides until the ingredients are just combined.
Transfer the dough to a well-floured surface and knead it 6 times, just until it comes together into a 6-inch round. The dough will still be sticky. (See Notes.)
Cut a large X in the top of the round and place the loaf of a baking sheet. Dust it with 1 teaspoon of flour and bake it for 30-35 minutes until golden brown.
Transfer the loaf to a rack and cool it completely before cutting it into slices. (See Notes.)
When ready to assemble the grilled cheese, spread 4 slices of the soda bread with the prepared basil pesto, and then add a few slices of the Irish cheddar. Form sandwiches by topping off each slathered side with a second piece of bread, and then butter both exterior sides of the bread. (See Notes.)
Heat a skillet over medium heat and transfer the prepared sandwiches to the pan, searing them just until golden on one side. Flip the sandwiches once to brown the second side and continue melting the cheese. If necessary, cover the pan to ensure the cheese will melt. Cut the grilled cheese and serve warm.
Every time I make this bread I find it gets better and better with the less you knead it. Many soda breads are often not kneaded at all prior to baking.
The loaf will provide more than 8 slices of bread, but I like to use the leftovers to make homemade soda bread croutons, which I toss with a bit of olive oil and fresh garlic.
I've become a bit of a grilled cheese connoisseur over the years and have found that you get much more even browning if you butter the bread first, rather than place the prepared sandwiches in a pan of melted butter.