Preheat oven to 325°F (162°C). Pat short ribs dry and season all over with 1 tbs kosher salt and 1 tsp black pepper. Heat 2 tbs olive oil in a heavy Dutch oven over medium-high heat. Sear short ribs on all sides until deep brown, about 3–4 minutes per side. Remove ribs and set aside. Add diced onion, carrot, and celery to the pot and cook until softened, 6–8 minutes. Add 4 minced garlic cloves and 2 tbs tomato paste; cook 1–2 minutes until fragrant. Deglaze with 1 cup red wine, scraping browned bits. Add short ribs back to pot, pour in 3 cups beef stock (or enough to come halfway up the ribs), add bay leaf and thyme, bring to a simmer. Cover and transfer to oven; braise until meat is fork-tender, about 2.5–3 hours.

Shred ragù

Remove short ribs from braising liquid and let rest until cool enough to handle. Reserve 1/2 cup of strained braising liquid and discard bones and bay leaf. Shred meat finely with two forks or chop finely; mix with a few tablespoons of reserved braising liquid to moisten. Taste and adjust seasoning with a pinch of salt and pepper. Cool to room temperature, then refrigerate until firm, at least 1 hour (this makes forming arancini easier).

Prepare risotto

In a heavy saucepan, melt 3 tbs butter over medium heat. Add arborio rice and stir to coat for 1–2 minutes until edges are translucent. Pour in 1/2 cup dry white wine and stir until mostly absorbed. Add warm (kept on low heat) beef stock or water, 1/2 cup at a time, stirring frequently and allowing each addition to be nearly absorbed before adding more; continue until rice is creamy and al dente, about 18–20 minutes total. Stir in 3/4 cup grated parmesan, 3/4 cup whole milk, 1 tsp salt, and 1/2 tsp black pepper; adjust seasoning. The risotto should be thick and coolable — if too loose, cook a minute longer. Spread risotto on a shallow tray to cool quickly to room temperature, then chill until firm, about 1 hour.

Filling & Coating

Assemble arancini

Set up an assembly station with flour in one shallow bowl, beaten eggs mixed with 2 tbs cold water in a second bowl, and breadcrumbs in a third bowl (mix panko and fine breadcrumbs if using). With lightly oiled hands, scoop about 2 tablespoons of chilled risotto into your palm and flatten into a disc. Place about 1 teaspoon of shredded short rib ragù and one small cube of cheddar in the center. Encase the filling with more risotto, shaping into a compact 1.5–2 inch ball, fully sealing the filling. Continue until all risotto is used, yielding about 16–20 arancini.

Coat arancini

Toss each rice ball in flour, shaking off excess, then dip into the egg wash, and finally roll in breadcrumbs to coat completely. For extra-crispy texture, double-coat: return to egg wash and breadcrumbs again. Place coated arancini on a tray and refrigerate 15–20 minutes to set the crust.

Fry arancini

Heat vegetable oil in a deep pot to 350°F (175°C). Fry arancini in batches—do not overcrowd—for 3–4 minutes, turning occasionally, until deep golden brown and heated through. Use a slotted spoon to transfer to a paper towel-lined tray and season lightly with a pinch of salt. Keep warm in a 200°F (93°C) oven while frying remaining batches.

Vodka Cream Sauce & Garnish

Make vodka cream sauce

In a skillet over medium heat, melt 2 tbs butter and sauté 1/2 cup minced shallot until translucent, 2–3 minutes. Add 1 minced garlic clove and cook 30 seconds. Stir in 1 tbs tomato paste and cook 1 minute. Carefully add 1/4 cup vodka and let it simmer 2–3 minutes to reduce slightly and cook off alcohol. Add 1/2 cup reserved braising liquid (or beef stock) and 1 cup heavy cream; bring to a gentle simmer. Stir in 1/4 tsp red pepper flakes, 1/2 tsp salt, and 1/4 tsp black pepper. Simmer until sauce thickens slightly, about 5–7 minutes, stirring occasionally. Taste and adjust seasoning. If sauce is too thick, thin with up to 2 tbsp reserved stock or water. Keep warm.

Serve

Spoon a generous amount of vodka cream sauce onto warm plates. Place 3–4 arancini on each plate atop the sauce. Sprinkle shaved parmesan over the arancini and finish with a small pinch of freshly ground black pepper. Serve immediately so arancini remain crisp.