A correct way to initialize provider

[bomb-on]

Hi!
I'm trying to use EIP6963 provider provided by my browser along with ether's BrowserProvider but I'm facing an issue of missing either request() or getSigner() methods, depending on which provider I'm using.

If I use EIP6963 provider directly then I get an error when I try to use provider.getSigner() method, while provider.request() works fine:

// const provider = selectedEIP6963Provider.provider;
// const signer = await provider.getSigner();

Uncaught (in promise) TypeError: provider.getSigner is not a function
    at connectToSelectedWallet (index.html:42:37)

If I wrap EIP6963 provider in BrowserProvider then I get an error when I try to use provider.request(), while provider.getSigner() works fine:

// const provider = new ethers.BrowserProvider(selectedEIP6963Provider.provider);
// const accounts = await provider.request({ method: 'eth_accounts' });

Failed to connect to MetaMask: TypeError: provider.request is not a function
    at connectToSelectedWallet (index.html:37:24)
    at HTMLSelectElement.<anonymous> (index.html:51:5)

Is there a way to create a provider with ethers to have both functions available all the time?