Step 1

Sprinkle beef with salt and pepper. Melt butter in large wide pot over medium-high heat. Working in 2 batches, cook beef until browned, about 6 minutes per batch. Transfer to large rimmed baking sheet. Add celery, carrots, and onion to pot and sauté until beginning to soften and brown, stirring often, about 5 minutes. Add wine, broth, Sherry, garlic, bay leaves, and thyme sprig; bring to boil, scraping up browned bits. Season with salt and pepper. Return ribs to pot, propping up on sides and arranging in single layer. Cover, reduce heat to medium-low, and simmer 1 hour.

Step 2

Using tongs, turn ribs over in pot. Cover and simmer until ribs are tender and sauce is very thick, occasionally rearranging ribs in pot to prevent sticking, about 1 1/2 hours longer. Uncover and cool 30 minutes.

Step 3

Transfer ribs to work surface. Discard bay leaves and thyme sprig. Spoon off fat from sauce in pot. Remove meat from bones; discard bones. Cut meat into 3/4- to 1-inch pieces, trimming any fat. Return meat to pot. DO AHEAD Can be made 2 days ahead. Chill until cold; cover and keep chilled. Rewarm just until lukewarm before using.

Step 4

Melt butter in large skillet over medium-high heat. Add onions, sprinkle with salt, and sauté until beginning to brown, stirring frequently, about 10 minutes. Add vinegar and sugar and cook until almost all vinegar is absorbed, about 1 minute. Season with salt and pepper. Transfer to microwave-safe bowl; cool. DO AHEAD Can be made 2 days ahead. Cover; chill. Microwave in 15-second intervals until lukewarm before using.

Step 5

Line 2 large rimmed baking sheets with waxed paper. Butter 8 bread slices; place 4 slices, buttered side down, on each prepared sheet. Divide short rib mixture among bread slices, about 1/2 cup for each. Divide cheese among sandwiches. Spoon about 1/4 cup onions over each sandwich. Place large handful of arugula atop onions. Top with remaining 8 bread slices. Spread bread with butter. DO AHEAD Can be prepared 1 hour ahead. Cover with plastic wrap and store at room temperature.

Step 6

Heat griddle or 2 large skillets over medium heat. Working in batches, cook sandwiches until bread is golden brown and cheese melts, about 3 minutes per side. Transfer to work surface. Cut each in half on diagonal. Transfer to plates and serve.