Preheat oven to 350°.
Remove white papery skin from garlic head (do not peel or separate the cloves). Wrap garlic head and each shallot separately in foil. Bake at 350° for 1 hour; cool 10 minutes. Separate cloves; squeeze to extract garlic pulp. Discard skins.
Place 1 tablespoon butter in a microwave-safe bowl. Microwave at HIGH 15 seconds or until melted. Rub butter over pork; sprinkle with 1 teaspoon thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Heat a large ovenproof nonstick skillet over medium-high heat. Add pork; cook 5 minutes, browning on all sides. Place pan in oven; bake at 350° for 20 minutes or until a thermometer registers 160° (slightly pink). Remove pork from pan; cover with foil. Let stand 10 minutes.
While pork stands, melt remaining 1 tablespoon butter in pan over medium-high heat. Add shallots and garlic; sauté 2 minutes. Stir in potato; sauté 2 minutes. Stir in mushrooms, remaining 1/2 teaspoon salt, and remaining 1/4 teaspoon pepper; sauté 2 minutes. Stir in broth and wine; bring to a boil. Cover, reduce heat, and simmer 6 minutes or until the potatoes are tender. Stir in spinach, arugula, and remaining 1/4 teaspoon thyme; cook 1 minute or until greens are lightly wilted. Slice pork crosswise into 12 pieces. Serve pork over potato mixture.
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