Heat cast-iron skillet over high for 2 minutes. Add olive oil, shallots, scallion whites and thyme. Lower heat to medium; cook, stirring occasionally, until shallots are golden and crispy on edges, 5 minutes. Transfer mixture to paper towels. In bowl, mix together cheese and cream.
Stir in cooked shallot mixture, red onion and scallion greens; season. Cover and refrigerate until ready to serve. (Can be made up to 2 days ahead:)