Winter Kin Sorcerous Origin
Your sorcerous power comes from a mysterious time in your life, when you disappeared into a blizzard, the ice broke on a frozen lake, or you went alone into the snow and your tracks just... stopped. The next thing you know, some time - anywhere from a day to a few years - has passed, you haven't aged the same amount of time (if at all), and you have a strange command over the frost and creatures of winter. Or maybe you just have a bloodline connection to the Unseelie Court or a divine power of Winter.
Cloak of Winter
At 1st level, you gain resistance to cold. As a reaction after taking cold damage from any source, you gain immunity to cold until the beginning of your next turn. Armor of ice and snow briefly takes shape over your clothing and any armor you wear; this does not restrict you in any way.
You ignore the effects of environmental cold, frigid water, slippery ice, and thin ice. Terrain covered in ice or snow is never difficult terrain for you.
Winter Birthright
At 1st level, when you deal cold damage to a creature, you may use a bonus action to blast the target with gale-force winds as well. Make a Shove attack against the creature, using your spell attack bonus in place of a Strength (Athletics) check.
When you would deal cold damage to a creature that naturally (not as part of a spell or other temporary effect) has resistance or immunity to cold damage, the creature must pass a Wisdom saving throw against your spell DC or become either charmed or frightened, at your discretion, until the beginning of your next turn. On a success, there is no further effect.
Icy Transformation
Starting at 6th level, when you spend any sorcery points as part of casting a spell that deals damage, you may change the spell's damage type to cold.
Further, when your current sorcery points are less than half of your maximum sorcery points, you may choose to have vulnerability to fire damage. If you do, all ranged attacks against you have disadvantage, and any creature that deals damage to you with a melee attack also takes one point of cold damage for each level of sorcerer you possess. This effect continues until your current sorcery points are equal to or greater than half your maximum, or you choose to end it, whichever comes first.
Biting Cold
Starting at 14th level, add your Charisma bonus to the damage of any spell you cast that deals cold damage. If you cast a spell that makes multiple separate attacks, only add your Charisma bonus to the damage of a single attack.
Master of the Frost
Starting at 18th level, you assert your absolute mastery over winter. When you deal cold damage to a creature, you may spend a bonus action to force it to make a Constitution saving throw or be restrained. If it fails this saving throw by 5 or more, it is instead paralyzed. If it fails this saving throw by 10 or more, it is instead petrified (turned entirely into ice).
When you use your Winter Birthright feature to impose the charmed or frightened condition upon a creature resistant or immune to cold, the duration extends to 1 minute. If its Challenge Rating is less than your sorcerer level, you may instead cast dominate monster on the creature without expending a spell slot.
Design Notes
First off: yes, I have created a subclass to support one of the Disney characters most intensely adored by girls under the age of... let's say ten. This does not bother me in the slightest, and if it helps some D&D-playing parent get a child interested in this hobby, then I will call that a solid win. (It's also a fair rendition of a particular Adventure Time character. Also fine, for the same reason.) I wasn't thinking of either of those characters when I first came up with this class, though.