Preheat oven to 375F.
In a bowl, whisk the eggs until smooth and frothy. Add dill and whisk through to combine. Set aside.
Preheat a non-stick pan over medium-high heat. Add olive oil and heat until shimmering. Add mushrooms and cook, undisturbed, until golden brown, 3 to 4 minutes. Set aside ¼ cup of mushrooms for garnish.
Add green beans and cook, stirring occasionally, until slightly softened, around 3 minutes. Add broccoli, season with a pinch of salt, and cook, stirring occasionally, until slightly softened, around 3 minutes.
Using a spatula, push the vegetables to one side of the pan. If the pan is dry, add 1 tablespoon olive oil. In the empty side of the pan, add the onions, red bell pepper, and chili pepper. Season with a pinch of salt and cook, stirring often, until the onions are soft and translucent.
Add ghee or butter, if using, and season with pepper. Toss or stir everything to combine and evenly distribute the vegetables across the pan. Slowly pour in the eggs so as not to disturb the vegetables.
Cook the eggs for 2 minutes or until you can gently shake the pan and notice that the eggs have started to peel away from the bottom. Lay in the slices of tomato and cheese and transfer to oven. Cook the frittata for 15 minutes or until the centre is slightly firm and beginning to set.
Remove the frittata from the oven and transfer it to a serving platter. Garnish with reserved mushrooms, fresh dill and serve with Roasted Garlic Aioli.