The sun is studied in depth, and the affect of it's cycles on climate has been greatly researched and debated. The hypothesis that the sun is causing global warming has been considered and proved false.
Several researchers have noted a correlation between solar activity, ie the number of sunspots, and climate. There still seems to be some debate as to how close that correlation is and how much difference the sun has made in past climates -- but it seems likely that cycles of the sun have had great impacts on earth's climate. BUT, the correlation clearly ended in the 1970s. The earth's temperatures have accelerated while solar activity has declined. If solar activity does have a significant impact on earth's temperatures, then something is overwhelming that effect and causing warming when we would naturally be having cooling. If that is true, and something else is overwhelming the solar minimum, then we can expect warming to further accelerate as the sun cycles back up to a more active status. There is nothing about the sun that would be causing warming right now, but it may be that more warming is coming if the sun cycles back up to an average state.
Secondly, the stratosphere is cooling rather than warming. If the warming was due to increased energy from outside of our atmosphere, then we would have warming at the highest altitudes as well as at the surface. But if the warming is coming from and enhanced greenhouse effect, then more radiation would be reflected back to the surface rather than reaching the top altitudes and we would see warming at the surface and cooling at the top. Well, that's been tested and the cooling in the stratosphere again disproves the solar hypothesis and supports the enhanced greenhouse effect.
Your analogy is over-simple, but yes if the toast burns the first suspician is that the toaster has been turned up. But once you test the toaster and you learn that it is not heating more -- in fact is heating a bit less -- and still the toast is burning, you can eliminate the toaster as the cause and look for other causes. If you know that you started buying different bread (comparable to changing the chemical composition of our atmosphere), you then start suspecting the bread and start looking for other ways to test that. If all your other tests (toasting in other toasters, etc) keep supporting that it's the bread you become very sure that it is indeed the bread.