In a bowl with an electric mixer beat together cheeses, garlic paste, and chile morita paste or minced chipolte chiles. Transfer mixture to a small serving bowl and chill, covered, at least 4 hours or overnight.
In a small saucepan combine chile morita paste or minced chiles chipoltes, piloncillo, and cream and cook over low heat, stirring occasionally and breaking up sugar, until piloncillo is melted and sauce is smooth, about 20 minutes. Transfer sauce to a small serving bowl and chill, covered, for at least 4 hours or up to overnight.
Spread cheese on grilled bread, corn chips or crackers and lightly drizzle with sauce.