After careful configuration, your settings should work for both occasions.
The reason to have a thin pellet (cake) when you fill your double basket with a single shot amount of grounds is, the area of the double basket is set to hold twice as much grounds of a single shot basket.
The reason behind that is the Darcy equation which defines the filtering properties based on
- area (constant in both baskets),
- viscosity (constant for water),
- filtration velocity (constant, if we assume you have constant pressure),
- filtration permeability (we'll come to that later),
- and the thickness of the pellet (we'll come to that later).
So, non-constants in this equation are the thickness of your pellet and the filtration permeability. Let's consider permeability first. Kozeny&Carman equation defines this for porous medium. I can say, in short, reliable grind size defines this. Therefore, if you are ensured that you have homogeneously ground coffee and tamp them with equal force (say 15 kg-f), you can safely assume permeability as constant, too.
Now, the last one: thickness of the pellet. Please see the below photograph I have taken, first.
As you see, the single spout portafilter has less space and the water flows from a narrower set of holes to keep the thickness constant. When you fill the double spout portafilter, the water flows uniformly and the thickness will still be the same.
So, all you need to do is setting your grinder to single shot amount (7-9 grams). Then, using single or double portafilter according to your needs. When you need a single shot, fill your portafilter with a single shot amount of coffee. Filling your double portafilter with double the amount ends up a doppio or two espressos in two cups.