Rub each filet with 1 teaspoon olive oil. Evenly season on all sides with garlic salt, pepper, and paprika, rubbing spices into filets. Let filets sit at room temperature for 30 minutes.
In a nonstick sauté pan, heat remaining 1 teaspoon olive oil over medium-high heat. Add meat to pan, and sear filets on all sides until browned, 2 to 3 minutes per side. Transfer filets to prepared baking sheet.
Bake filets until desired degree of doneness is reached**, 5 to 7 minutes for rare. Add more baking time for well done.
When filets are cooked, top each piece with ½ teaspoon butter. Wrap securely in foil. Let rest for at least 15 minutes.
Using a 2½-inch round cutter, cut 16 rounds from sourdough bread slices and 8 rounds from wheat bread slices. Using the same cutter, cut 8 rounds from cheese slices.
Fifteen minutes prior to serving, slice filets into 4-x-¼-inch pieces. Set aside.
Spread Horseradish Aïoli on one side of each sourdough round and one side of each wheat round.
Top aïoli side of 8 sourdough rounds evenly with ¼ cup arugula leaves. Ruffle 2 filet slices to fit each bread round. Top each with a wheat round, aïoli side up. Top evenly with remaining ¼ cup arugula. Top each with a cheese slice and 3 tomato slices. Top each with a sourdough round, aïoli side down. Secure with a decorative pick.
Serve immediately.
Notes
*We used Penzeys Hungarian Half-Sharp Paprika (penzeys.com).
†We used Pepperidge Farm Farmhouse Style Sourdough Bread.
‡We used Pepperidge Farm Farmhouse Style Wheat Sandwich Bread.
**The best way to make sure meat reaches a safe internal temperature is to use a meat thermometer. Filets should be cooked to at least 140° (rare).
MAKE-AHEAD TIP: Prepare filets mignons and Horseradish Aïoli, and refrigerate in airtight containers for no more than a day. Cut out bread rounds, and store at room temperature in airtight containers for up to a day.
Recipe by TeaTime Magazine at https://teatimemagazine.com/beef-cheddar-triple-stack-sandwiches-recipe/