Spicy Beef and Mushroom Udon with Broccolini

Name: Spicy Beef and Mushroom Udon with Broccolini
Ingredients: 2 teaspoons neutral cooking oil; 1 pound ground beef; 4 teaspoons garlic (minced); 2 teaspoons soy sauce; 1 teaspoon sesame oil; 1 teaspoon chili oil (optional); Crushed red pepper (to taste); 4 ounces shiitake mushrooms (stems discarded and caps thinly sliced); 1 teaspoon ginger (minced); 5 cups chicken stock; 2 teaspoons rice vinegar; 1 bunch broccolini (woody ends trimmed); 16 ounces fresh or frozen udon; 4 eggs; 2 teaspoons distilled white vinegar; Salt and pepper to taste; Sesame seeds (optional, for garnish); Thinly sliced daikon radish (optional, for garnish)