Melt 1 tsp. butter with oil in a large nonstick frying pan over medium heat. Cook vegetables, stirring, until barely tender, about 5 minutes. Stir in tarragon, season to taste with salt and pepper, and scoop out of pan into a bowl.
Butter 1 side of bread slices with remaining butter. Put 2 slices in pan, buttered side down, and top with cooked vegetables. Layer with cheeses and sprinkle with pepper to taste. Set remaining bread slices, buttered side up, on top. Cook sandwiches, turning once, until golden brown and cheese is melting, about 8 minutes.
Note: Nutritional analysis is per sandwich.