Step 1Cut each tomato into lengthwise quarters and combine them in a bowl with the tarragon.
Step 2Melt the half-cup of butter in a small saucepan over medium heat. Add the shallots and cook until they soften and become fragrant. Remove the butter from the heat and let cool 2 or 3 minutes. Lightly season the fillets with salt and pepper.
Step 3When the butter is cooler, but still hot enough to soften the tomatoes but not so hot as to cook them through, pour it over the tomatoes and stir to combine. Stir gently to avoid smashing the tomatoes. Season to taste with salt (it will take about 1 teaspoon). If the tomatoes lack acidity, add a few drops of red wine vinegar. Set the sauce aside while cooking the fish.
Step 4Add 1 tablespoon of oil and 1 tablespoon of butter to a large nonstick pan and heat over high heat. Place one to two fillets in the pan depending on the size and cook until golden brown, 2 to 3 minutes. Turn the fish over and cook 1 to 2 minutes until done. Add the remaining oil to the pan as needed to finish cooking the remaining fish.
Step 5Place a fillet on each serving plate and spoon a portion of the tomato butter (about one-third cup) over the top. Serve immediately.