This looks like a great question for a coding interview ;) You can achieve this using template literal types that were introduced quite a while ago. Here's how:
type ExtractByDotNotation<TObject, TPath extends string> =
// If TKey has a dot, split it into two parts: before the dot and after the dot
TPath extends `${infer TKey}.${infer TRest}` ?
// Checking if the key exists in the object
TKey extends keyof TObject ?
// Get type recursively
ExtractByDotNotation<TObject[TKey], TRest> :
// Key provided is invalid
never :
// If the path doesn't contain a dot, use it as a key
TPath extends keyof TObject ?
TObject[TPath] :
never
Playground link
You can simplify the first clause using infer ... extends ... from TS 4.7
type ExtractByDotNotation<TObject, TPath extends string> =
// Constraining TKey so we don't need to check if its keyof TObject
TPath extends `${infer TKey extends keyof TObject & string}.${infer TRest}` ?
ExtractByDotNotation<TObject[TKey], TRest> :
TPath extends keyof TObject ?
TObject[TPath] :
never
Playground link
However, I wouldn't recommend using this type in production code, as there may be uncertainties in how the template string part behaves with multiple dots in TPath. TS documentation doesn't specify if TKey will be before the first dot, last dot, or somewhere random in between.