In a large pot, low heat, pour the olive oil. Add the garlic and onion. Then, the 3 different peppers. Toss well.
Then, add the tomatoes, the eggplant and the zucchinis. Stir gently with a wooden spoon.
Let it simmer at low heat for 20 minutes, half covered with a lid or until the vegetables are cooked but not mushy.
Add salt and the Herbes de Provence.
With about half of the leftover:
Place in a large pot or bowl and puree with a hand mixer until smooth.
In a pan loaf, cut the lasagna sheet to fit the pan. Spread with your lasagna puree. Spread half a cup of goat cheese. Cover with a lasagna sheet, spread the puree, the other half of goat cheese. Another lasagna sheet, the rest of the puree and all the mozzarella on top.
Bake in the oven at 375C for 25-30 minutes.