Typescript typing: function arguments go through keys of nested object

Question

interface TestObject {
    person: {
        firstName: string;
        lastName: string;
        address: {
            country: string;
            city: string;
            street: string;
        }
    }
    id: string;
}
const t: TestObject = {} as any;

type K<T> = T extends object ? keyof T : never;
function f<T, K1 extends K<T>>(object: T, key: K1): T[K1];
function f<T, K1 extends K<T>, K2 extends K<T[K1]>>(object: T, key1: K1, key2: K2): T[K1][K2];
function f<T, K1 extends K<T>, K2 extends K<T[K1]>, K3 extends K<T[K1][K2]>>(object: T, key1: K1, key2: K2, key3: K3): T[K1][K2][K3];
// ...
function f() {
    return 1 as any;
} 

const result1 = f(t, 'person', 'address');
/*
const result1: {
    country: string;
    city: string;
    street: string;
}
*/

const result2 = f(t, 'id', '');
/*
Argument of type 'string' is not assignable to parameter of type 'never'.ts(2345)
*/

Can the same be done without "death by a thousand overloads"? If there was something like Tail<T> type alias for const tuples, i think i know how i could do it.