Demi-glace Omlet with Rice

I first made demi-glace (brown sauce made of beef stock and vegetables) in a quicker way and it wasn't that easy! This recipe needs to improve but will write down the best I could...Maybe if I used a pressure cooker, that may be a solution...

Demi-glace (brown sauce)
50 - 60 grams thinly sliced beef
1 tbs olive oil
1/8 cup flour
1 tbs tomato paste or tomato sauce
1/4 cup red wine (or white)
3/4 - 1 cup beef broth (I substituted with oofu-dashi and real beef)
1.5 tbs Worcestershire sauce
1 bay leaf
Some thyme
Some nutmeg
*1 tbs & 2 tsp usuta sauce (Japanese version of Worcestershire sauce)
*1 tbs ketchup or tomato paste
*2 tsp sugar

Without the last three ingredients, the sauce may be too thin. By adjusting the last three ingredients, you can make the sauce richer or less rich.

1. Sear beef in a saute pan with oil and take out beef to put it aside.
2. Cook flour in the saute pan until it turns brown and add tomato paste and cook until it thickens.
3. Pour wine over the saute pan and let it evaporate.
4. Stir in beef broth (I put back my seared beef) and add onions (and mushrooms, if you have), Worcestershire sauce, bay leave, thyme and nutmeg.
5. Simmer them over medium-low heat until the sauce thickens.
6. Add the last three ingredients to your preference.

Will update this recipe later including the omlet & rice part. Hoping I can try other quicker way to make this sauce!