The Green Transition to net-zero carbon emissions in Europe requires massive financing efforts, with estimates of 620 billion EUR annually, but the headwinds are substantive. Central banks seem overstretched and busy tightening to combat inflation; treasuries are subject to austerity-inducing fiscal rules; and banking systems are afflicted by non-performing loans, fragmentation, and risk aversion. We employ the framework of ‘monetary architecture’ to analyse the EU’s monetary and financial system as a constantly evolving hierarchical web of interlocking balance sheets and study its capacity to find ‘elasticity space’ to meet the financing challenge. To this end, we draw on a four-step scheme for green macro-financial governance along the financial cycle of balance sheet expansion, funding, and final contraction. We find that, first, Europe’s monetary architecture still has ample elasticity space to provide a green initial expansion due to its developed ecosystem of national, subnational, and supranational off-balance-sheet fiscal agencies. Second, as mechanisms lack to consciously organise the distribution of long-term debt instruments across different segments, its capacity to provide long-term funding is limited. Third, institutional transformation in the last two decades have greatly improved the capacity of the European monetary architecture to counteract financial instability by providing emergency elasticity. Fourth, the capacity of the European monetary architecture to manage a final contraction of balance sheets is limited, which is a general quandary in modern credit money systems. Our analysis points to the need for further investigations into techniques for monetary architectures to manage long-term funding and balance sheet contractions.