Is there a Boolean formula that is resistant to bias from a polytime adversary?
There does not exist such a family of circuits. Assume to the contrary that $\{\phi_k\}_{k \in \mathbb N}$ was such a family. 

We will now show the strategy that achieves bias at least $\frac{1}{20k}$ for $k$ big enough, by referring to Impagliazzo's Theorem 4.7.2 on page 71 of http://cseweb.ucsd.edu/~russell/format.ps : 

Let $\mathcal A$ be the strategy that:

1) Picks a bit string uniformly from $x \in \{0,1\}^k$, and picks a number $ i \in \{0,\ldots,2k-1\}$ uniformly. Now, $i = 2j+b$, for $b \in \{0,1\}$ for an unique $j$ and $b$.

2) $\mathcal A$ should send the first $j$ messages in correspondance with the first $j$ coordinates of $x$. 

3) If $b=0$ Then $\mathcal A$ should wait for answer from the other party. Otherwise, it should just proceed:

4) $\mathcal A$ picks a new bit string $(x'_{j+1}, \ldots x'_k) \in \{ 0,1 \}^{k-j}$ uniformly, and then calculate 2 estimates of the Expected value of $\phi$, calculated by making polynomially many uniform samples from $\{0,1\}^{k-j-b}$ corresponding to the remaining conversation of the honest party:


$E_1$ : Estimate of expected value where preceding conversation is fixed, and  $(x_{j+1}, \ldots x_k)$ is fixed.
$E_2$ : Estimate of expected value where preceding conversation is fixed, and  $(x'_{j+1}, \ldots x'_k)$ is fixed.


5) The bit string corresponding to the highest expected value, is used for the rest of the conversation.

If assuming perfect estimation of expected value, we achieve at least $\frac{1}{16k}$ bias, and since there is some (exponentially decreasing error) caused by the estimation (which can be bounded by additive Chernoff-Hoeffding bound), we just say a bias of at least $\frac{1}{20k}$ for $k$ big enough.

For now, I will just refer to Impagliazzo's proof for the analysis. It is not the easiest proof to read, if one is not used to his style of writing, but it is correct, and I can explain specific details, if anybody have interest.