Season short ribs with salt and pepper. Sear in a 5-6 quart-size lightly oiled oven-proof pan; remove ribs from pan.
Add onion, carrot, celery and garlic; cook stirring frequently 3-5 minutes until vegetable are softened. Add tomato paste and stir for 1-2 minutes.
Add red wine; bring to a simmer and cook until slightly reduced.
Return ribs to pan. Add beef stock to cover the ribs. Cover pan and bake 2 to 2 1/2 hours until ribs are tender. Turn ribs half-way through cooking time. If liquid has reduced to less than half, add additional stock.
Remove ribs from liquid. Strain and press liquid through a large-hole sieve, pressing most of the solids through the sieve. Measure liquid and add additional stock if needed to make about 3 cups.
Shred short rib meat. There should be about 6 cups (1-1/2 pounds)
Directions for Potato Hash
Lightly coat a large non-stick skillet with cooking oil. Place over medium heat; add cubed potatoes, bell pepper, onion, garlic, and herbs. Cook stirring frequently until potatoes are cooked al dente, about 15 minutes.
Add shredded short rib meat and cooking liquid to potatoes and vegetables. Simmer over medium heat until liquid is absorbed and potatoes are just tender. Potatoes should remain in cubes.
Serve portions of hash topped with two eggs. Garnish with sour cream and giardiniera.