Cook udon noodles according to package directions. Drain and set aside.
In a large pot, melt butter over medium heat. Add flour and whisk constantly for 2-3 minutes to create a roux.
Add curry powder to the roux and cook for 1 minute until fragrant.
Gradually add broth while whisking to prevent lumps from forming.
Add sliced onion and carrots to the pot. Simmer for 10 minutes until vegetables begin to soften.
Add sliced meat and cook for 5-7 minutes until cooked through.
Stir in soy sauce, mirin, and honey. Simmer for 5 more minutes until sauce thickens.
Add cooked udon noodles to the curry broth and gently combine.
Serve hot, garnished with sliced green onions.