Heat a 12-inch non-stick skillet in medium heat. Add 1-tablespoon vegetable oil. Allow it to get really hot.
Add the chunks of mushrooms to the hot pan ensuring that each piece has enough room to cook. Cook mushrooms until they loose some of their juices, 6-8 minutes. At the very end, add 1-tablespoon butter and ensure that all mushrooms are coated (glazed) with butter. Transfer mushrooms in a medium bowl and set aside.
Crack eggs in a large bowl. Add milk, salt and pepper and whisk until thoroughly incorporated.
Add half of the cooked mushrooms into the egg mixture. Using a spatula, give it quick stir.
Pour the egg and mushroom mixture back in the skillet (you can use the same skillet or a smaller skillet) ensuring that the mushrooms are evenly distributed in the pan. Put a lid on and allow it to cook in medium heat until edges are set, 5-6 minutes. This will make a slightly wet frittata in the middle. If you want more of a dry frittata, cook an additional minute or two.
In the mean time, make the lemon vinaigrette: Whisk lemon juice, lemon zest, shallot, olive oil, salt and pepper in a small bowl. Drizzle it over the arugula. Mix to ensure that all the arugula leaves are coated with the vinaigrette. Taste for seasoning and add if necessary.
Using a spatula, slide the frittata to a plate. Place the rest of the Portobello mushrooms in the middle of the frittata.
Using your fingers, divide the goat cheese into small chunks and scatter them on the frittata, around the mushrooms.
Place the arugula in the middle.
Serve immediately.