We investigate the growth trajectory of online platforms operating as market makers in two-sided matching markets. Their expansion, often referred to as the ``chicken-and-egg dilemma'' in the literature, poses an inherent challenge: service providers find value in joining the platform because of the presence of customers, while customers are drawn to the platform by the availability of service providers. Much of the existing literature addresses this expansion challenge through either a static market-clearing lens or aggregate dynamic models, emphasizing how network externalities shape a platform's growth trajectory and equilibrium size. These approaches do not fully capture how gradual growth must be implemented through server-level admission, assignment, and compensation decisions. Embedded in this growth process are intertemporal participation constraints that the platform must satisfy to attract and retain forward-looking servers. To explicitly account for the stochastic dynamics of a platform's gradual growth, we model the system as a Markov process. Potential servers and customers arrive according to Poisson processes, and customer demand is non-decreasing in the number of active servers. Under mild conditions, we characterize the platform's optimal admission, assignment, and compensation policies. The platform admits servers at the maximum feasible rate until it reaches an endogenous target size, which can be smaller than the static market-clearing optimum. In the baseline model, seniority-based compensation acts as an operational substitute for financially subsidized growth: instead of using upfront subsidies to induce early server participation, the platform uses future cash flows to reward early servers for joining when demand is limited and for helping the platform reach later, higher-demand states.

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