I looked at the logo first (before reading what you wrote) and said to myself "There's no way leoparded pizza clipart exists"
I'm guessing the leoparding took some time?
The logo feels a bit busy. You have a lot of information to convey, so I understand that it can't be that simple, but, with everything going on, it feels a little cluttered.
The black cheese shadow squiggles on the pizza have to go. It's the focal center of the logo and right now, it just looks hairy. The leoparding looks great, but the center of the pie needs work. I like the multi-colored stonework, but I think, for the sake of a cleaner look, you might want to try one color for the stone.
The Flirting with Fire text looks a bit isolated. As you look at the logo, your eye tends to zero in on 'Mobile Wood Fired Pizza' more than 'Flirting with Fire.' The company name should have a more central focus.
I like the font, but I think, somehow, a better font might be out there. This is fire-y, but it adds to the busy-ness and also could impact legibility. As the artistry of the font increases, it can be difficult to maintain legibility.
I'm just throwing this out there, but a chimney might give you a stronger, shield like aesthetic and could convey that it's an oven a tiny bit faster. Maybe.