Melt butter in a 10- to 12-inch frying pan over medium-high heat.
Add mushrooms, garlic, and basil; stir often until all liquid has evaporated and mushrooms begin to brown, 10 to 12 minutes.
Add cheese and milk; stir until cheese melts and sauce begins to boil.
Meanwhile, in a 5- to 6-quart pan, bring 4 quarts water to boiling over high heat.
Add frozen tortellini and cook, uncovered, until just barely tender to bite, 15 to 20 minutes.
Drain and pour into a bowl; add mushroom sauce and mix well.
Spoon tortellini and sauce equally onto 4 to 6 heated dinner plates; garnish with parsley sprigs.