In a medium bowl, combine shredded cheddar cheese, shredded Monterey Jack cheese, grated Parmesan cheese, 1/2 cup of mayonnaise, and crumbled dried tarragon leaves.

Mix the cheese mixture until evenly combined.

Divide the cheese mixture evenly among 4 slices of bread, spreading to the edges.

Top with the remaining 4 slices of bread to form sandwiches.

Gently but firmly press the sandwiches.

In a wide, shallow bowl, whisk together the eggs, milk, the remaining 1/4 cup of mayonnaise, dried thyme leaves, and kosher salt.

Mix the egg mixture until smooth.

Preheat oven to 325°F (163°C).

Dip each sandwich into the egg mixture, coating both sides.

Heat 1 tablespoon of olive oil in a 12- to 14-inch nonstick saute pan until smoking.

Add 1 tablespoon of butter to the pan and heat until the sizzling subsides.

Place 2 sandwiches in the pan and cook slowly until golden brown on the first side, about 3 minutes.

Flip each sandwich and cook on the other side for 3 more minutes.

Remove the sandwiches from the pan to a baking pan.

Repeat the cooking process with the remaining oil, butter, and sandwiches.

Place the baking pan with the sandwiches in the preheated oven.

Bake for 10 minutes.

Serve hot.