Preheat oven 400 degrees. Pat dry tuna, rub with olive oil, sprinkle evenly with kosher salt. Sear on the grill or pan on med-high heat for 2 -3 minutes on each side. Remove from heat and set aside for 3 minutes.
In medium bowl, combine; mayo, thyme, shallots, white pepper. Combine mayo mixture and tuna. With fork, flake tuna to desired texture. Spread butter evenly on one side of bread. Place 2 slices of bread, butter side down on a cookie sheet. Reserving half, sprinkle with cheddar cheese. Spoon tuna mixture evenly over both slices. Use the rest of the cheese on top of the tuna and top with bread, butter side facing you.
Using a weighted lid, panini skillet lid or another clean oven proof pan. Place it on top of your sandwiches. And put in oven for 5 minutes. Take out and turn. Remember the pan/lid on top is hot. Put it back in 5 minutes. Slice in halves and serve hot.
Find any recipe, add your own, make grocery lists easily from recipes. Enter any 3 ingredients and BigOven will tell you what you can make. It's free! (4.5/5 stars)