In a medium saucepan, combine the potatoes and garlic. Add 2 inches of water to the pan. Salt to taste liberal, bring to a boil over high heat, then lower the heat to a simmer.
Cook for about 20 minutes, or until the potatoes are soft.
After draining the potatoes, save 1 cup of the cooking water. Use a potato masher to return the potatoes and garlic to the saucepan.
Stir in 1/3 cup of olive oil and 1/2 cup of the cooking water. To make the potatoes spreadable, add additional water as necessary while stirring.
Add the chives and season with salt and pepper to taste. Mash potatoes in a covered dish and set aside.
In a food processor, pulse the mushrooms until they are roughly chopped. In a medium skillet over medium-high heat, warm the 2 tablespoons of olive oil until shimmering.
Cook for about 8 minutes, or until all of the moisture from the mushrooms has drained, after adding the mushrooms, 1 teaspoon salt, and several grinds of pepper. The tomato paste and rosemary should be added, and they should be cooked for about 3 minutes while stirring occasionally.
Add the peas and carrots after stirring in the liquid, and boil for 4 minutes.
With the use of a spoon or spatula, spread the potatoes into an even layer after removing from the heat and dousing them with mashed potatoes.
Bake for about 25 minutes, or until the top is golden and the edges are bubbling.