Subspace codes, and in particular cyclic subspace codes, have gained significant attention in recent years due to their applications in error correction for random network coding. In this paper, we introduce a new technique for constructing cyclic subspace codes with large cardinality and prescribed minimum distance. Using this new method, we provide new constructions of cyclic subspace codes in the Grassmannian $\mathcal{G}_q(n,k)$ of all $k$-dimensional $\mathbb{F}_q$-subspaces of an $n$-dimensional vector space over $\mathbb{F}_q$, when $k\mid n$ and $n/k$ is a composite number, with minimum distance $2k-2$ and large size. We prove that the resulting codes have sizes larger than those obtained from previously known constructions with the same parameters. Furthermore, we show that our constructions of cyclic subspace codes asymptotically reach the Johnson type bound II for infinite values of $n/k$.