To prepare potatoes, place potatoes in a saucepan; cover with water. Bring to a boil; reduce heat, and simmer 20 minutes or until tender. Drain. Return potatoes to pan; add milk, 1/4 teaspoon salt, and 1/4 teaspoon black pepper. Mash potato mixture with a potato masher to desired consistency. Stir in cheese. Keep warm.
To prepare crawfish, melt butter in a large nonstick skillet over medium-high heat. Add onion and garlic; saute 5 minutes. Stir in corn; saute 2 minutes. Add 1/2 teaspoon salt, paprika, red pepper, thyme, 1/4 teaspoon black pepper, and crawfish; saute 5 minutes or until thoroughly heated. Remove from heat; stir in green onions. Serve crawfish mixture over mashed potatoes. Garnish with thyme sprigs, if desired.