To make the brown butter, place the butter in a small saucepan over medium heat and immediately begin whisking. Stir as it bubbles, and after 3 to 4 minutes (or sooner!) you should see brown bits appear on the bottom of the pan. Remove from heat immediately and continue to whisk for another 30 seconds or use. Pour the butter into a bowl and and stick the bowl in the fridge or the freezer so it comes to a solid, spreadable form. You can do this ahead of time and just leave the butter sit at room temperature.
Pat the tomatoes dry with a paper towel, especially if they are extra juicy. Do the same with the burrata cheese since it was most likely soaking in water or brine.
Heat a large skillet or grilled over medium-low heat. The key is cooking the sandwiches low and slow. Sprinkle garlic salt and pepper all over the tomato slices. Pull apart the balls of burrata so you have 4 pieces total. On the bread slices, layer one piece of pulled-apart burrata cheese, then layer tomatoes on top. Cover with the other piece of burrata. Spread the outsides of the bread with the room-temp brown butter.
Cook the sandwich until the outsides of the bread are golden brown and crunchy and the cheese is melty. If at any time it seems like you need more butter, add it or add a drizzle of olive oil to the pan. Remove from the skillet and sprinkle with parmesan cheese. Slice and serve immediately with fresh basil.
Note: the key here is to use a thick and sturdy, even grainy bread. If your bread is very soft (such as white bread, hawaiian bread, etc), the cheese and tomatoes will cause it to be very soggy.