In a large pot of boiling, salted water, combine the potatoes and turnips. Cover; cook until tender, 15 minutes. Drain; reserve 1 cup cooking liquid.
Meanwhile, in a small saucepan, heat the EVOO over medium heat. Working in batches, add the shallots; cook until crisp, 3 to 4 minutes. Drain on paper towels. Strain the cooking oil and reserve.
Mash the veggies with the reserved cooking liquid and 5 tbsp. of the shallot oil. Season with 1 tsp. salt and 1/4 tsp. pepper. Spoon into a serving bowl; top with the fried shallots.