How to Cook St-Hubert À La King Sauce?

I have trouble cooking the St-Hubert à la King Sauce, though I was able to cook it right with another stove and saucepan.

The issue I have is that the sauce does not get thick enough. Moreover, the sauce is a bit burned at the bottom.

Sometimes, the sauce, at the end, is almost as light as milk.

Another issue is that the sauce will often "grow" when it is too hot (this was not happening with the other stove) and I have to move the saucepan away (so that it wont spill) and/or lower the temperature. (The sauce boils with my current stove, but it did not happen with the other stove.)

I almost constantly stir the sauce with a whisk and I progressively lower the temperature.

With the other stove, I kind of feel with the whisk when it is the right time to lower the temperature, but I don't know why I don't feel it with my current stove.

I tried with a saucepan with a thicker bottom, but it does not really help.

Sometimes, I add carrots and celeries with the other ingredients at the start.

So, what can I do to get a thick sauce without problem?