Whisk together sugar, cocoa powder, cinnamon, and salt in a saucepan; stir in boiling water. Whisk until sugar is dissolved.
Bring cocoa mixture to a simmer over medium-high heat, stirring constantly, about 2 minutes; stir milk and 1/2 cup half-and-half into water mixture. Cook and stir just until hot, about 2 minutes. Remove saucepan from heat; stir in remaining half-and-half and vanilla extract.
Divide cocoa into mugs, top with marshmallows, and serve.
